Mr. Ahmadinejad said he sent all three letters, which offered long philosophical musings and governing advice, via mail to the care of the Swiss ambassador in Tehran, via message to Mr. Trump’s Twitter account and to a White House email address. The Swiss Embassy looks after American interests in Iran in the absence of direct diplomatic ties.

It was unclear if Mr. Trump ever received the letters. White House officials said they needed more information about precisely how and when they were sent but pointed out that Mr. Ahmadinejad could not have directly messaged them via Twitter because Mr. Trump does not follow him. The Swiss Embassy in Tehran declined through a spokesman to comment.

Mr. Ahmadinejad said he had not been reprimanded for attempting to correspond with Mr. Trump and that Mr. Khamenei can change his mind and approve negotiations with Washington if the administration shifts its approach. He pointed out that Mr. Khamenei, who has the final word on Iran’s relations with the United States, had allowed nuclear talks with the Americans under President Barack Obama.

The timing of the messages of both Mr. Ahmadinejad and Mr. Zarif were notable: The Trump administration has sent several signals in recent days that it wants to begin talks with Iran with “no preconditions.”

And for the first time since Mr. Trump abandoned the nuclear agreement, both sides are talking about the need to negotiate, even if each has set out unilateral demands that the other must meet.

There is no guarantee, of course, that both sides will find a way. Mr. Khamenei has described Mr. Trump as an evil trickster and has prohibited talks with him under any circumstances. And Gen. Qassem Suleimani, the commander of the elite Quds Force of the Revolutionary Guards who is rumored to be a future president, has said that talking to Washington would be like surrender.

Mr. Ahmadinejad conceded that for talks to happen, in his view, the United States would need to soften its approach.