Mueller will accept written answers from Trump on questions about whether his campaign conspired with Russia's election interference, Mueller's office told the president's lawyers in a letter, two people briefed on it said.
The report is based on a letter Mueller and his investigators gave to Trump's legal team Friday.
Special counsel Robert Mueller sent a letter to Trump's legal team expressing a willingness to accept written answers to some questions, a source says.
The special counsel's office would decide whether to proceed with an interview after receiving the written answers from Trump.
Following the "kissing" comments, the President went off on a familiar diatribe about the supposed illegitimacy of Mueller's investigation, which has already resulted in almost 40 indictments against Trump associates and Russians.
Trump's personal attorneys, Rudy Giuliani and Jay Sekulow, declined to comment.
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Sources said the tone prompted Trump allies to believe that any interview will be more limited in scope than the president's legal team initially believed. In March, Mueller raised the prospect of subpoenaing Trump if he refused to be questioned voluntarily.
The president claimed he had a previous business dispute with Mueller, but he offered no specifics - and then claimed to have compromising photographs of the two former Federal Bureau of Investigation directors together.
The letter was first reported by the New York Times. The person said the Trump legal team has not responded to the offer.
The president made the freaky claims during a wide-ranging interview with the conservative Daily Caller website, whose correspondent asked Trump whether Mueller was interfering in the midterm elections by not ending his Russian Federation probe by an artificial September 1 deadline.
Mueller so far has not requested to ask any questions about obstruction, but included a line in his message to Trump's attorneys that he may want to ask questions about obstruction at some point in the future. That's an effort that could lead to a Supreme Court challenge by the White House.
And for another, Trump claimed, without any proof that Comey and Mueller are "best friends" and even more, of the "hugging and kissing" variety. "You know, he's Comey's best friend", said Trump.