Mueller raised the issue of interviewing Trump during a late-December meeting with the president's lawyers John Dowd and Jay Sekulow. Mueller deputy James Quarles, who oversees the White House portion of the special counsel investigation, also attended.
The special counsel's team could interview Trump soon on some limited portion of questions — possibly within the next several weeks, according to a person close to the president, who spoke on the condition of anonymity to describe internal conversations.
Scenes from Trump’s second six months in office
"This is moving faster than anyone really realizes," the person said. Trump is comfortable participating in an interview and believes it would put to rest questions about whether his campaign coordinated with Russia in the 2016 election, the person added.
However, the president's attorneys are reluctant to let him sit for open-ended, face-to-face questioning without clear parameters, according to two people familiar with the discussions. Since the December meeting, they have discussed whether the president could provide written answers to some of the questions from Mueller's investigators, as President Ronald Reagan did during the Iran-contra investigation. They have also discussed the obligation of Mueller's team to demonstrate that it could not obtain the information it seeks without interviewing the president.
The legal team's internal discussions about how to respond to a request for an interview were first reported Monday morning by NBC News.
Dowd and Sekulow declined to comment.
In a statement, Ty Cobb, the White House lawyer overseeing the administration's response to the Mueller investigation, said that "the White House does not comment on communications with the OSC out of respect for the OSC and its process," referring to the special counsel's office.
"The White House is continuing its full cooperation with the OSC in order to facilitate the earliest possible resolution," Cobb added.
Cobb had repeatedly said all interviews of White House personnel by Mueller's office were on schedule to be completed by the end of December or early this year. On Monday, he said he remains confident that any portion of the investigation related to the president or the White House will wrap up shortly.
Mueller and Trump's legal team plan to meet again soon to discuss both the possible terms and substance of the interview, as well as Mueller's timeline for the investigation, according to one person familiar with the plan.
Trump's lawyers hope to obtain from the special counsel's team a clear idea of the categories of questions that would be posed to the president.
For months, Trump's legal team has been researching the conditions under which the president would be required to submit to an interview with the special counsel, who is investigating Russia's meddling in the 2016 election.
"No lawyer just volunteers their client without thinking this through," said one of the people familiar with the talks.
It has long been expected that Mueller would seek to interview Trump, in part because the special counsel is scrutinizing whether actions he took in office were attempts to blunt the Russia investigation, according to people familiar with questions posed to witnesses.
The president also dictated a misleading statement later released by his eldest son, Donald Trump Jr., about a meeting that Trump Jr. had with a Russian lawyer during the presidential campaign.
Veteran prosecutors said it is unlikely that Mueller would agree to have any witness, even the president, submit a declaration or provide written answers to questions to avoid a sit-down interview.
Some experts said a presidential interview could signal that Mueller's investigation into Trump's actions is nearing its end, but they cautioned that the special counsel might have a different strategy.
"It would certainly seem they would be close to wrapping up as it relates to the core matter they are investigating," said Solomon Wisenberg, a deputy independent counsel who questioned President Bill Clinton in 1998. "You would want to know as much as possible before you go to the president. "
Asked on Saturday if he had agreed to be interviewed by Mueller, Trump said he had nothing to hide.
"Just so you understand, there's been no collusion, there's been no crime, and in theory everybody tells me I'm not under investigation. Maybe Hillary [Clinton] is, I don't know, but I'm not," Trump told reporters at Camp David. "But we have been very open. We could have done it two ways. We could have been very closed, and it would have taken years. But you know, sort of like when you've done nothing wrong, let's be open and get it over with."
"Because, honestly, it's very, very bad for our country," the president added. "It's making our country look foolish. And this is a country that I don't want looking foolish. And it's not going to look foolish as long as I'm here."
In June, after Comey told a congressional panel that Trump had privately asked for his loyalty, the president said he would be willing to testify under oath to dispute the fired FBI director's claims.
"One hundred percent," Trump said when asked if he would give a sworn statement to Mueller.
Sitting presidents have been interviewed by prosecutors in the past, though courts have urged government investigators to seek such interviews only when they cannot obtain relevant information another way.
Clinton's attorneys repeatedly fought independent counsel Kenneth W. Starr's attempts to interview their client until investigators obtained a subpoena to force his testimony. It was the first grand jury subpoena served on a sitting president. Clinton then negotiated to testify before a grand jury via video and audio link from the White House Map Room.
In the videotaped interview in August 1998, which lasted four hours and saw questions from three prosecutors, Clinton admitted to inappropriate sexual activity with White House intern Monica Lewinsky, but he claimed he had been legally correct in denying that he had had sexual relations with her. He also denied having committed perjury in a lawsuit brought by Paula Jones.
Not all presidential interviews with prosecutors have come at the end of an investigation.
In 2004, President George W. Bush sat for an in-person interview with special prosecutor Patrick J. Fitzgerald, who was investigating whether senior White House aides leaked a CIA operative's identity and broke her cover as punishment for her husband's criticism of the Iraq War. Bush volunteered for the interview, which lasted 70 minutes and was conducted in the Oval Office. Bush was far from the last one interviewed in the probe; Fitzgerald later questioned several more central witnesses.
"The leaking of classified information is a very serious matter," White House press secretary Scott McClellan said at the time, adding that Bush was "pleased to do his part" to aid the investigation.
Reagan testified in the Iran-contra investigation while in office and twice more after he left office. He also answered in writing some written questions presented to him by the grand jury and the independent counsel in the probe.
In 1975, President Gerald R. Ford was interviewed as part of a grand jury investigation into an assassination attempt. In a taped session in the Old Executive Office Building, Ford shared his recollections of events when Lynette "Squeaky" Fromme, a Charles Manson follower, tried to shoot Ford at close range in Sacramento in September 1975. The tape was used at her trial that year.
Rosalind S. Helderman contributed to this report.