WASHINGTON — President Trump, the author of "The Art of the Deal," has been projecting his usual bravado in public this week about the prospects of repealing the Affordable Care Act. Privately he is grappling with rare bouts of self-doubt.

Mr. Trump has told four people close to him that he regrets going along with Speaker Paul D. Ryan's plan to push a health care overhaul before unveiling a tax cut proposal more politically palatable to Republicans.

He said ruefully this week that he should have done tax reform first when it became clear that the quick-hit health care victory he had hoped for was not going to materialize on Thursday, the seventh anniversary of the act's passage, when the legislation was scheduled for a vote.

Two of his most influential advisers — Stephen K. Bannon, his chief strategist, and Gary D. Cohn, the National Economic Council director, who had a major role in pushing the bill — came to agree, and did not like the compromise that was emerging. So on Thursday night, Mr. Trump delivered an ultimatum.

He dispatched his budget adviser, Mick Mulvaney, to a conference of House Republicans and told them they had to vote on Friday. And if the bill fails, he said, Mr. Trump will move on.

A president who prefers unilateral executive action and takes intense pride in his ability to cut deals finds himself in a humbling negotiation unlike any other in his career, pinned between moderates who believe the health care measure is too harsh, and a larger group of fiscal conservatives adept at using their leverage to scuttle big deals cut by other Republican leaders.

Over the years, Mr. Trump has proved to be a resilient operator, and even his most scathing critics do not rule out his ability to pull off some kind of a deal, even at a late hour.

"I don't know whether he will ultimately succeed or fail, but I will tell you that President Trump is so transactional, who knows what transactions he will be willing to make to pass this," said Representative Nancy Pelosi, the Democratic leader, who passed the Affordable Care Act in 2010 as speaker.

"So far he's acting like a rookie. It's really been amateur hour," she added. "He seems to think that a charm offensive or a threat will work — that saying 'I can do this for you' or 'I can do this against you' will work. That's not the way it works. You have to build real consensus, and you have to gain a real knowledge of the policy — and the president hasn't done either of those things."