The Washington Post, citing two people familiar with Mueller's plans, said the special counsel was seeking to question Trump about his ousting of FBI director James Comey and national security advisor Michael Flynn.
They told the newspaper that Trump's attorneys have worked out terms for the president's interview with Mueller's team which could be presented to the special counsel "as soon as next week".
"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," said the Post.
A day earlier, the US Justice Department had confirmed that Attorney General Jeff Sessions was questioned by Mueller on his investigation on potential links between the Trump campaign and Russia in the presidential election.
Mueller has already interviewed several of the close aides and family members of Trump over the issue.
Trump, earlier, had hinted that it "seems unlikely" that he would allow Mueller to "interview" him.
Meanwhile, the White House said it supports "full transparency" around a secret memo criticising the FBI but it was for the House Intelligence Committee to decide whether to release it or not.
"We certainly support full transparency, and we believe that's with the House Intel Committee to make a choice," White House press secretary Sarah Sanders told reporters at her daily news conference.
She reiterated that the White House would be cooperative with Mueller.
"As we've said probably just about every day this year, that we're going to be fully cooperative with the special counsel and we're going to continue to do that throughout the process," Sanders said when asked whether the president was open to being interviewed by Mueller.
"But we're also not going to comment on who may or may not, or could be interviewed at any point," she said.
Sanders said Trump wants the controversy should come to an end.
"The President wants to see this end, and he wants to see them finally come to the same conclusion that I think most in America have, that there is nothing to this.
"They've spent the better part a year looking, digging, obsessing over trying to find something and are yet to find anything," she said.
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