At the same time, though, they have determined that for Mr. Mueller to complete his inquiry in a timely manner, Mr. Trump will need to sit down for questioning. Mr. Giuliani plans to try to work with Mr. Mueller to come up with a way to question Mr. Trump that both sides are comfortable with.

The addition of Mr. Giuliani comes at a particularly tumultuous time for the president. Last month the president’s lead lawyer, John Dowd, quit the team after he determined the president was not following his advice. For much of the past month, the team has been led by Mr. Sekulow, who has had to assemble a new group of lawyers to deal with the issues in New York and another team to confront Mr. Mueller.

At the same time, Mr. Mueller has been pressuring Mr. Trump to sit down for an interview. The president, who initially said he was eager to answer Mr. Mueller’s questions under oath, is said to be more skeptical of an interview in the wake of the raid on Mr. Cohen’s office.

After Mr. Dowd quit, many well-known lawyers turned down the opportunity to join the president’s legal team. Some said that they did not believe Mr. Trump would listen to them and that their firms did not want to be associated with the president. But after the raid on Mr. Cohen, many more lawyers have become interested in working for Mr. Trump, according to people briefed on the matter. The lawyers believe the government overstepped its bounds by executing a warrant at a lawyer’s office and have contended the government violated the attorney-client privilege between Mr. Cohen and his clients.

Mr. Trump negotiated the discussions to have Mr. Giuliani join his team with Mr. Giuliani directly, a person close to the process said. Mr. Trump had repeatedly offered Mr. Giuliani the job of attorney general during the transition, but Mr. Giuliani turned it down because he wanted to be secretary of state.

Mr. Trump turned to Senator Jeff Sessions, Republican of Alabama, to be attorney general, but has publicly criticized Mr. Sessions for recusing himself from the Russia inquiry. Mr. Trump has said that Mr. Sessions should be protecting him from the inquiry.

Some close to the president believe he could try to replace Mr. Sessions with Mr. Giuliani in the coming months, although Mr. Giuliani would face an extremely difficult confirmation hearing in the Senate. When Mr. Giuliani sought the secretary of state job, Trump advisers, including the president’s son-in-law, Jared Kushner, raised concerns about his business dealings and paid speeches to a shadowy Iranian opposition group that until 2012 was on the State Department’s list of foreign terrorist organizations.