Discussions about a Trump interview come amid the broader inquiry into Russia's interference in the 2016 election, a wide-ranging investigation that has already led to charges against four former Trump advisers.
Mueller now appears to be turning his attention to Trump and key witnesses in his inner circle, raising the pressure on the White House as the administration enters its second year.
Last week, Attorney General Jeff Sessions was interviewed for several hours by special counsel investigators, according to Justice Department officials. He is the first member of Trump's Cabinet to be questioned in the probe.
Months ago, the special counsel's office also briefly interviewed Comey, who at the time vouched for the contents of memos he wrote about private conversations he had with the president, according to people familiar with the matter. The Sessions and Comey interviews were first reported by the New York Times.
Trump's attorneys have crafted some negotiating terms for the president's interview with Mueller's team, and they could be presented to the special counsel as soon as next week, according to the two people.
The president's legal team hopes to provide Trump's testimony in a hybrid form — answering some questions in a face-to-face interview and others in a written statement.
A spokesman for the special counsel's office, Peter Carr, declined to comment. A White House spokesman referred questions to the president's legal team. Two attorneys for Trump, Jay Sekulow and John Dowd, declined to comment.
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. In 1998, President Bill Clinton testified for more than four hours before a grand jury via a video link after being subpoenaed by independent counsel Kenneth W. Starr.
Within the past two weeks, the special counsel's office has indicated to the White House that the central subjects investigators wish to discuss with the president are the departures of Flynn and Comey and the events surrounding their firings.
Mueller has also expressed interest in Trump's efforts to remove Sessions as attorney general or pressure him into quitting, according to a person familiar with the probe. The person said the special counsel was seeking to determine whether there was a "pattern" of behavior by the president.
Flynn resigned last February after The Washington Post reported that he had misled Vice President Pence and other administration officials about his communications with Sergey Kislyak, then the Russian ambassador to the United States.
Late last year, Flynn pleaded guilty to lying to the FBI about his contacts with Kislyak. Trump then tweeted that "he had to fire General Flynn because he lied to the Vice President and the FBI." Previously, the White House had cited only the false statements to Pence as a rationale for dismissing Flynn.
Trump fired Comey in May, several days after the then-FBI director told Congress that he could not comment on whether there was evidence of collusion between the Trump campaign and Russia. At the time, Comey was overseeing the Russia probe. Comey later testified that the president had asked him several months earlier whether he could see a way to "letting Flynn go."
Earlier this month, Trump declined to say whether he would grant an interview to Mueller and his team, deflecting questions on the topic by saying there had been "no collusion" between his campaign and Russia during the 2016 presidential election.
"We'll see what happens," Trump said when asked directly about meeting with the special counsel.
Behind the scenes, the president has told his team of lawyers that he is not worried about being interviewed because he has done nothing wrong, according to people familiar with his views. His attorneys also support a sit-down, as long as there are clear parameters and topics.
However, some of Trump's close advisers and friends fear that a face-to-face interview with Mueller could put the president in legal jeopardy. A central worry, they say, is Trump's lack of precision in his speech and his penchant for hyperbole.
People close to Trump have tried to warn him for months that Mueller is a "killer," in the words of one associate, noting that the special counsel has shown interest in the president's actions.
Roger Stone, a longtime informal adviser to Trump, said he should try to avoid an interview at all costs, saying that agreeing to such a session would be a "suicide mission."
"I find it to be a death wish. Why would you walk into a perjury trap?" Stone said. "The president would be very poorly advised to give Mueller an interview."
Sessions, who has recused himself from oversight of the special counsel investigation, could be a key witness to the events under scrutiny. In 2016, he met at least twice with Kislyak. After Trump was elected, Sessions was one of a small number of administration officials involved in discussions with the president that led to the firing of Comey.
Sessions's lawyer, Chuck Cooper, who accompanied him to his special counsel interview last week, declined to comment.
The attorney general's role in the investigation and his supervision of the Justice Department have been marked by controversy. At times, he has struggled to explain what was said in private meetings that are now of interest to investigators.
During his confirmation hearing in early 2017, Sessions was asked what he would do if he learned that there had been contacts between Russians and the Trump campaign. He answered: "I did not have communications with the Russians.''
After The Washington Post reported that he met at least twice with Kislyak in 2016, Sessions announced that he was recusing himself from investigations involving the election, based on the advice of Justice Department ethics lawyers.
He has since maintained that he misunderstood the scope of the question at his confirmation hearing, and that his meetings with Kislyak were fleeting or strictly in his capacity as a U.S. senator. In announcing his recusal, Sessions said: "I never had meetings with Russian operatives or Russian intermediaries about the Trump campaign."
That assertion is contradicted by the accounts Kislyak provided to his superiors in Moscow, according to current and former U.S. officials.
Kislyak reported to his bosses that he discussed campaign-related matters, including policy issues important to Moscow, with Sessions during the 2016 presidential race.
At the time, Sessions was a top foreign policy adviser to candidate Trump. Kislyak's accounts of the conversations were intercepted by U.S. spy agencies, which regularly monitor the communications of senior Russian officials in the United States and Russia.
One U.S. official said Sessions has provided "misleading" statements that are "contradicted by other evidence." A former official said the intelligence indicates that Sessions and Kislyak had "substantive" discussions on matters including Trump's positions on Russia-related issues and prospects for U.S.-Russia relations in a Trump administration.
Sessions's recusal from the Russia probe has continued to rankle the president, according to administration officials, and the attorney general has become embroiled in other internal battles in recent months.
In December, Sessions pushed FBI Director Christopher A. Wray to remove and replace some of his top aides, particularly Deputy Director Andrew McCabe. Trump and others have argued that those top aides, who served at the FBI under Comey, are biased against the president.
Devlin Barrett and Ellen Nakashima contributed to this report.