Rudolph W. Giuliani, the new head of President Trump’s legal team, met with the special counsel’s office on Tuesday as he tries to expedite the end of the far-flung inquiry into possible collusion between the 2016 Trump campaign and Russian officials, three people familiar with the meeting said.

Mr. Giuliani declined to confirm whether a meeting took place. But the three people briefed on the sit-down — the first since the former New York City mayor joined the legal team last week — described it as a productive effort to reopen negotiations over a possible interview that the special counsel, Robert S. Mueller III, is seeking with the president.

News of the meeting was first reported by The Washington Post.

In an interview, Mr. Giuliani said that his goal in the coming days was to find out where Mr. Mueller stands, particularly as it relates to the credibility of James B. Comey, the F.B.I. director whom the president fired in April 2017. Mr. Comey has been interviewed by Mr. Mueller, and he has been deeply, personally critical of the president in his media tour promoting his new memoir.

“At this stage of an investigation, it’d be highly unusual to let an ordinary client testify,” Mr. Giuliani said. However, he added, “This isn’t an ordinary client. This is the president of the United States.”