score:14
The book Manhattan, the Army and the atomic bomb has a chapter dedicated to the choice of bombing targets on pages 528 β 530. It makes clear that the original intention was to drop four bombs. Originally the targets selected were Kokura, Hiroshima, Niigata and Kyoto. Later it was decided to spare Kyoto for its historical relevance and bomb Nagasaki instead. Judging by this book, no additional bombings were planned. In particular, on page 541 it says:
[General Groves and General Marshall] agreed that, in view of Stimson's policy of using the bomb only to end the war, shipment of materials for a third bomb should be delayed until 13 August.
So there was no plan to use nuclear weapons in Operation Downfall. The atomic bombs were rather used to end the war before Operation Downfall becomes necessary. That matches what I know from other sources as well.
Upvote:5
Professor Alex Wellerstein examined this question on "Restricted Data: The Nuclear Secrecy Blog" in a post titled "The Third Shot and Beyond" (April 25, 2012). He presented evidence that the U.S. military was contemplating (not necessarily planning) use of atomic bombs in the invasion of Japan. There were seven more bombs expected to be available by the end of October 1945, that is, in time for the scheduled November invasion. The United States at that time was able to produce up to three atomic bombs per month.
Upvote:11
According to this document from the National Security Archive Electronic Briefing Book No. 162 at George Washington University they were planning on seven more bombs. As for targets however General John Hull states in the document (dated August 13 1945):
What General Marshall wants to know is the status of the development of these bombs now so we can best determine how to use them.
On finding that one bomb is ready to ship out Hull states:
...Two of them have had a tremendous affect on the Japanese as far as capitulation is concerned. the next one won't be effective in that respect. In other words, the Japs will or will not.
After further discussion of delivery expectations for other bombs and the use of them with American troops nearby, Hull concludes by saying he would like General Groves slant on how and when to use them. From this document it appears that while they were planning on building more bombs for use against Japan, they had not yet decided where to use them.