In contrast, Mr. Mnuchin, a former Goldman Sachs banker and Hollywood film producer, tends to be the Trump administration’s voice of moderation, willing to compromise to reach a deal. Last month, Mr. Mnuchin emerged from a meeting in Ms. Pelosi’s office to tell reporters that the White House was not looking for a shutdown fight or flirting with a debt limit default. Mr. Mulvaney stood, red-faced, to his left, waiting for a chance to interject.

When the Treasury secretary stopped speaking, the acting chief of staff jumped in, charging Ms. Pelosi with having raised her spending demands and asserting — incorrectly — that she had promised to raise the debt ceiling no matter what, a shift, he insisted, from her previous position.

“I shouldn’t even be engaged in a conversation that has him mischaracterizing,” Ms. Pelosi later said.

By no means has Mr. Mulvaney been completely shut out: A White House official said that on Tuesday, Mr. Mulvaney participated in a meeting with the president about the debt ceiling. And two administration officials said that Mr. Trump’s lack of interest in the talks meant that the acting chief of staff could dip in and out of the discussions, sticking to the big picture without getting sucked into the minutiae.

At times, he has privately expressed bemusement at what he calls Mr. Mnuchin’s naïveté when it comes to haggling with Congress. In the spring, when Mr. Mnuchin suggested that House Democrats would be amenable to a “clean” raising of the debt ceiling — without preconditions, like raising the spending caps — Mr. Mulvaney insisted that Ms. Pelosi would surely try to extract concessions — as he would have.

He is likely to be proved right.

And Mr. Mulvaney certainly sees himself in charge. At a Heritage Foundation forum on Wednesday, Mr. Mulvaney said that he often joked that the White House budget office was now under the conservative think tank’s management.

But two administration officials acknowledged that Mr. Mulvaney’s diminished role in the budget negotiations could be in response to the bipartisan contingency on Capitol Hill that does not want him there. Mr. Mulvaney was seen poorly by those legislators, White House officials suggested, because they want to spend more than he does.