They are convinced that Mr. Mueller lacks the legal standing to question Mr. Trump about some of the matters he is investigating, like the president’s role in providing a misleading response last summer to a New York Times article about a meeting Mr. Trump’s son Donald Trump Jr. had with Russians offering dirt on Hillary Clinton. The advisers have also argued that on other matters — like the allegations that the president asked the former F.B.I. director, James B. Comey, to end the investigation into the former national security adviser Michael T. Flynn — the president acted within his constitutional authority and cannot be questioned about acts that were legal.
One of the few voices arguing for cooperating with Mr. Mueller is Ty Cobb, the White House lawyer whom Mr. Trump also brought on to deal with Mr. Mueller’s investigation. Since Mr. Cobb was hired in July, he has argued that the White House should do everything possible to cooperate with Mr. Mueller’s investigation.
But Mr. Dowd has taken the lead on dealing with the special counsel about an interview and has been discussing the matter with Mr. Mueller’s office since December.
Others close to Mr. Trump have also cautioned him against a freewheeling interview. Marc E. Kasowitz, the president’s longtime personal lawyer from New York who initially dealt with the special counsel after Mr. Mueller took over the Russia investigation last May, has also consistently said that the president should not agree to the interview.
Chris Christie, the former governor of New Jersey who led the presidential transition until just after the election, bluntly said last week that Mr. Trump should reject a request to be questioned.
“I don’t think the president of the United States, unless there are credible allegations — which I don’t believe there are — should be sitting across from a special counsel,” he said on ABC’s “Good Morning America.”
Newt Gingrich, the former House speaker and an informal adviser to Mr. Trump, echoed that advice.
“The idea of putting Trump in a room with five or six hardened, very clever lawyers, all of whom are trying to trick him and trap him, would be a very, very bad idea,” Mr. Gingrich said last month on “Fox and Friends.”
Presidents have often agreed to speak with federal prosecutors who are investigating their actions or those close to them. But Richard M. Nixon refused to turn over to the special prosecutor investigating him tapes of incriminating conversations with aides. The matter eventually went before the Supreme Court, which ruled in 1974 that the president, like every American, was not above the law and had to comply with the special prosecutor’s request.
“The upshot of the Nixon tapes case was that any president is going to have an extremely hard time resisting a request from a law enforcement officer,” said Neal K. Katyal, an acting solicitor general in the Obama administration and a partner at the law firm Hogan Lovells.
“In general,” he added, “presidents do sit for interviews or respond to requests from prosecutors because they take their constitutional responsibility to faithfully execute the laws seriously, and running away from a prosecutor isn’t consistent with faithfully executing the laws.”
Mr. Trump’s penchant for bravado has been a factor that his lawyers must contend with. The president has bragged to some aides that he would be able to clear himself if he talked to Mr. Mueller’s team.
“I’m looking forward to it, actually,” Mr. Trump told reporters at the White House last month, though he added, almost as an afterthought, that an interview would be “subject to my lawyers, and all of that.”
When pressed, he also said he would be willing to be questioned under oath. Questioning by Mr. Mueller would not be under oath, though lying to federal investigators is a crime.
Despite his penchant for exaggerations and falsehoods in public remarks, Mr. Trump has appeared far more aware of the legal risk of making false statements when he has been deposed in civil cases related to his businesses.
In one deposition related to a libel case that Mr. Trump brought against the journalist Tim O’Brien, Mr. Trump admitted more than two dozen times under oath that he had lied in the past about a range of subjects. Mr. Trump ultimately lost the case.
Since last summer, the White House has been in what Mr. Cobb has called “total cooperation mode.” He has issued few, if any, objections as Mr. Mueller’s team interviewed senior White House officials and reviewed reams of emails and other internal documents. The White House was operating on a strategy that, since Mr. Trump had nothing to hide, full cooperation was the best chance to bring the investigation to a quick, decisive end.
Mr. Cobb had told the president and the public that the Mueller inquiry would be over by the end of 2017, or soon after. But a month into 2018, it remains unclear when Mr. Mueller will wrap up the bulk of his work.
Privately, people close to the president have conceded that assuring Mr. Trump that the investigation would end by a certain date was primarily aimed at keeping him from antagonizing Mr. Mueller on his Twitter feed or in interviews.