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If you are denied entry in the US, two things can happen:
Note that the officer doesn't have to offer you to voluntarily depart; it's up to the officer, i.e. they can put you into removal proceedings (thus causing you a ban) even if you offer to depart on your own, or even if they at first offer to let you depart on your own, if you take too long to arrange your transportation, they can change their mind and put you into removal anyway. But they usually do offer you the option of voluntary departure unless your past immigration violations were particularly egregious.
(Note that the above only applies to ports of entry in the US. If you are denied entry at a US pre-clearance facility, e.g. if you are flying from Canada, Ireland, or Abu Dhabi, removal is not an option as you are not in the US. In that case, you would simply be left at the foreign airport and not be able to take your flight to the US.)