Robert S. Mueller III, the Justice Department’s special counsel who is leading a separate investigation into Russian meddling in the 2016 election, is also examining whether Mr. Trump tried to obstruct justice when he fired James B. Comey, the F.B.I. director who was running a federal inquiry into the matter.

Mr. Trump also called other lawmakers over the summer with requests that they push Mr. Burr to finish the inquiry, according to a Republican senator who requested anonymity to discuss his contact with the president.

This senator, who was alarmed upon hearing word of the president’s pleas, said Mr. Trump’s request to the other senators was clear: They should urge Mr. Burr to bring the Russia investigation to a close. The senator declined to reveal which colleagues Mr. Trump had contacted with the request.

Some of Mr. Trump’s advisers feared he would move to fire Mr. Mueller, an option that the president pointedly left open in an Oval Office interview with The New York Times in July.

During this time, Mr. Trump made several calls to senators without senior staff present, according to one West Wing official. According to senators and other Republicans familiar with the conversations, Mr. Trump would begin the talks on a different topic but eventually drift toward the Russia investigation.

In conversations with Mr. McConnell and Senator Bob Corker, the Tennessee Republican who is chairman of the Senate Foreign Relations Committee, Mr. Trump voiced sharp anger that congressional Republicans were not helping lift the cloud of suspicion over Russia, the senators told political allies. The Times reported in August that the president had complained to Mr. McConnell that he was failing to shield Mr. Trump from an ongoing Senate inquiry.

The earlier call with Mr. Burr, however, was perhaps the most invasive, given Mr. Burr’s role directly supervising the Senate’s investigation of Mr. Trump.