“No,” Mr. Trump said. “Nobody that I know of. Nobody.”

The White House has sought to portray Mr. Papadopoulos as an insignificant figure in the campaign.

Ty Cobb, the White House lawyer dealing with matters related to Mr. Mueller’s investigation, said the White House stood behind the president’s comments.

“The media’s willingness to inflate Papadopoulos, a young unpaid volunteer and supposed energy expert, into an important thought leader in the campaign or Russian operative is ludicrous,” Mr. Cobb said. “The evidence so far suggests he attended one meeting, said something about Russia and was immediately shut down by everyone in the room. It’s very important to remember that he is not a criminal now because of anything he did for the campaign — he is a criminal because he initially lied to the F.B.I.”

A Justice Department spokesman declined to comment.

Another member of the foreign policy team, Carter Page, said on Thursday that he told Mr. Sessions in passing in June 2016 that he planned to travel to Russia for a trip “completely unrelated” to his volunteer role in the campaign. “Understandably, it was as irrelevant then as it is now,” Mr. Page said. Mr. Page traveled twice to Russia in 2016.

Democrats in the Senate said on Thursday that they would push to have Mr. Sessions return to the Judiciary Committee for further questioning.

“He now needs to come back before the committee, in person, under oath, to explain why he cannot seem to provide truthful, complete answers to these important and relevant questions,” said Senator Patrick J. Leahy, Democrat of Vermont, who is on the Judiciary Committee.

Senator Richard Blumenthal of Connecticut, another Democrat on the committee, pointed out that Mr. Sessions’s testimony was under oath and “wasn’t just some random comment he made in passing on the street.”