What is the basis for the idea of a seven year peace treaty with Israel in the end times?

That a period of peace with the Nation of Israel will take place during the first three and one half years of the Great Tribulation, comes from Chapter 9 of the book of Daniel, and was verified by Jesus in chapter 24 of the book of Matthew.

Daniel 9:24 through 27 KJV

24 Seventy weeks are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy.

25 Know therefore and understand, that from the going forth of the commandment to restore and to build Jerusalem unto the Messiah the Prince shall be seven weeks, and threescore and two weeks: the street shall be built again, and the wall, even in troublous times.

26 And after threescore and two weeks shall Messiah be cut off, but not for himself: and the people of the prince that shall come shall destroy the city and the sanctuary; and the end thereof shall be with a flood, and unto the end of the war desolations are determined.

27 And he shall confirm the covenant with many for one week: and in the midst of the week he shall cause the sacrifice and the oblation to cease, and for the overspreading of abominations he shall make it desolate, even until the consummation, and that determined shall be poured upon the desolate.

The week of Daniel chapter 9 is actually a seven year period, and the midst of the week is three and one half years.

Jesus affirmed the prophesy of Daniel in Chapter 24 of the book of Matthew.

Mat 24:14 , 15, 21 and 22 KJV

14 And this gospel of the kingdom shall be preached in all the world for a witness unto all nations; and then shall the end come.

15 When ye therefore shall see the abomination of desolation, spoken of by Daniel the prophet, stand in the holy place, (whoso readeth, let him understand:)

21 For then shall be great tribulation, such as was not since the beginning of the world to this time, no, nor ever shall be.

22 And except those days should be shortened, there should no flesh be saved: but for the elect's sake those days shall be shortened.

The famous "70 Weeks" prophecy of Daniel 9 has been called by some Bible scholars
the "backbone" of Bible prophecy because it is the one that "seals up" other
prophecies. If this is true, it would seem to be rather important for those 
interested in the interpretation of Bible prophecy to figure it out correctly.
Daniel 9:24 mentions several things that are to be accomplished in the 70 Weeks:
"Seventy weeks are determined upon thy holy city, to finish the transgression, and   
to make an end of sins, and to make reconciliation for iniquity, and to bring in
everlasting righteousness, and to seal up the vision and prophecy, and to anoint
the most Holy."

Since this phrase "time, times, and a half" which comes from Daniel's prophecies 
is found in Revelation 12:14, it forms an obvious connection or link between the 
two books. This period is also given as 1,260 days in Revelation 12:6, 42 months 
in chapters 11 and 13. The 1,260 days would be 3.5 years, each year having 12
months of 30 days. In such a scheme, a year would have 360 days, although we know
today the length of the solar year is 365.25 days. Some have postulated there is
such a thing as a "prophetic year" that must be used for calculating the 70 Weeks.
An interesting interpretation of Daniel's 70 Weeks was proposed in 1895 by Sir
Robert Anderson, a brilliant biblical scholar, which made this assumption. Anderson
claimed that from the commandment to rebuild Jerusalem to the appearance of the 
Messiah (the seven weeks and threescore and two weeks of verse 25) worked out to 
exactly 483 biblical years equalling 360 days, or 173,880 days. 

In the solar-lunar calendar used by the Hebrews in ancient times, the length of a
month was determined by the phases of the moon. Our word "month" reflects the 
ancient connection with the moon's period. The precise duration of the moon's 
period is 29.531 days. It is close to 30 days, but not quite. Because of this, 
the ancient calendar which was used by the Babylonians, Hebrews, Greeks, and most
other nations in antiquity had years of varying length. Most years had 12 months
[or 'moons'], but some years had 13 months. The priests of the various nations 
would determine the length of the year by observation, and there was a fair bit of
confusion when bad weather interfered with observations of the moon.

The 70 weeks of Daniel are separated into three distinct parts, suggesting that
different units are involved in the various parts. [Or else, why would the 7 weeks,
and the 62 weeks, and one week be mentioned separately?] The emphasis is 
on "sevens," and while there may be 7-year periods involved, this may not be true
of all the parts. The three parts to the 70 Weeks prophecy resemble the three 
sections or parts in the phrase "time, times, and a half" that are mentioned in 
Daniel.

In Daniel 12:7, the phrase "time, times, and a half" is given as the duration of 
the period encompassed by his prophecies. The completion of this period, whatever 
it means, marks a time when "all these things shall be finished." The pattern which
is evident in the phrase "time, times, and a half" can be applied to two of Daniel's
mysterious time periods, the 1,290 days of Daniel 12:11, and the 1,335 days of Daniel
12:12, to help us understand and explain them. Looking at the pattern imbedded in
the phrase "time, times and a half," we can insert the different units for year 
lengths in the three divisions, as a kind of template, to obtain:

Daniel 12:7:                time  +   times   +  a half

Daniel 12:11:  1290 days =  390   +  2 x 360  +  360/2

Daniel 12:12:  1335 days =  360   +  2 x 390  +  390/2

John in Revelation applies this same pattern to derive his variations on three and
a half years: 

Revelation 11:2:    42 months =   12  +  2 x 12   +  12/2

Revelation 11:3:    1260 days =  360  +  2 x 360  +  360/2

Revelation 11:9:    3.5 days  =   1   +  2 x 1    +  1/2

Revelation 12:6:    1260 days =  360  +  2 x 360  +  360/2

Revelation 12:14:               time  +  times    +  a half

Revelation 13:5:    42 months =   12  +  2 x 12   +  12/2

A phrase quite similar to "a time, times and a half" is mentioned in Daniel 7:25;
here it is given as "a time, and times, and the dividing of time." Most Bible
scholars have assumed they are the same, but although the two phrases are similar, 
they are not identical. The pattern or structure here is similar to that of the 70
Weeks of Daniel 9:24-27, which has a period of 7 weeks, plus 62 weeks, plus a final
divided week.

Daniel 7:25:        time  +   times  + dividing of time

Daniel 9:24-27:   7 weeks + 62 weeks + divided week = 70 weeks

Now, what about the final, or 70th week? This final week is a figurative period of
seven years, and it is during this "week" that Christ "confirms the covenant with
many," as stated in Daniel 9:27. This period began when Christ began his ministry. 
His ministry lasted three and a half years. The prophecy said he would be "cut off,"
a reference to the crucifixion, after 62 weeks [Daniel 9:26].

In the final, 70th week, Christ is to "confirm the covenant with many." This final
week is still in effect, because Christ still "confirms the covenant" with 
believers. The first half of this week was the ministry of Christ on the earth. The
last half is symbolic of the entire time of the Church. 

There are two seemingly conflicting descriptions of the end times for Israel, that many resolve by figuring there will be a peace treaty. Since the tribulation is 7 years, they assume that to be the duration.

To wit, the references in question are:

1. Per Matthew 24:6, it will be a time of war

   You will hear of wars and rumors of wars, but see to it that you are not alarmed. Such things must happen, but the end is still to come.

2. Per Ezekiel 38, it will be a time of "peace" (although, see #3)

   7 “Be ready and keep ready, you and all your hosts that are assembled about you, and be a guard for them. 8 After many days you will be mustered. In the latter years you will go against the land that is restored from war, the land whose people were gathered from many peoples upon the mountains of Israel, which had been a continual waste. Its people were brought out from the peoples and now dwell securely, all of them.

3. The simplest resolution comes from continuing to read Ezekiel 38:

   9 You will advance, coming on like a storm. You will be like a cloud covering the land, you and all your hordes, and many peoples with you.

   10 “Thus says the Lord God: On that day, thoughts will come into your mind, and you will devise an evil scheme 11 and say, ‘I will go up against the land of unwalled villages. I will fall upon the quiet people who dwell securely, all of them dwelling without walls, and having no bars or gates,’ 12 to seize spoil and carry off plunder, to turn your hand against the waste places that are now inhabited, and the people who were gathered from the nations, who have acquired livestock and goods, who dwell at the center of the earth

   The idea is actually fairly simple - it is a treacherous time. There is all sorts of war, and israel is duped into a bad alliance. (Note: This is a recurring theme throughout the Prophets and Kings) The peace treaty with an unscrupulous nation (the Anti-Christ) gives the appearance of peace, but is actually a setup for a very bad outcome.

4. This also reads well with Daniel 9

   And he shall make a strong covenant with many for one week, and for half of the week he shall put an end to sacrifice and offering. And on the wing of abominations shall come one who makes desolate, until the decreed end is poured out on the desolator.”

   Traditionally, premillenial dispensationalists view the "70th week" of Daniel as the end times. The first sixty-nine weeks are viewed as the time between Daniel and the Messiah, then the Church age puts the 'pause' button on the 70th week. The tribulation is basically that last week.

This is the traditional pre-trib / dispensationalist reading, such as those popularized by Tim LaHaye in the Left Behind series. I am quite familiar with the reading, but remain unconvinced either way by it. As such, this answer is not based on the merits of the claim, only that the combination of the two is typically the basis for the claim.