Many of the potential questions relate to whether Mr. Trump obstructed justice, people briefed on the discussions have said. They have said they expect the interview to be completed by the end of February or early March.

Ty Cobb, the White House lawyer leading the response to the investigation, said Mr. Trump was speaking hurriedly and intended only to say that he was willing to meet.

“He’s ready to meet with them, but he’ll be guided by the advice of his personal counsel,” Mr. Cobb said. He said the arrangements were being worked out between Mr. Mueller’s team and the president’s personal lawyers.

The White House has made many witnesses available for interviews with prosecutors, and officials have said there are no discussions about Mr. Trump speaking before a grand jury, which is how prosecutors speak to witnesses under oath. Interviews with agents and prosecutors are not conducted under oath, but lying to the F.B.I. is a felony.

Pressed on whether he would be willing to answer questions under oath, Mr. Trump first asked a reporter whether Hillary Clinton, his 2016 campaign rival, had done so in the investigation into her use of a private email server while she served as secretary of state. Mrs. Clinton gave a voluntary interview to F.B.I. investigators in July 2016, and was not under oath, as is typical for such sessions.

President Bill Clinton testified under oath in 1998 about his relationship with a White House intern. He was questioned on camera in the White House Map Room, and the testimony was broadcast to a Washington grand jury room.

Mr. Trump spoke with reporters in the doorway of his chief of staff’s office, interrupting a background briefing with a senior administration official on immigration with his own roughly one-minute informal news conference before he departed for the World Economic Forum in Davos, Switzerland.