Two people briefed on the matter told The New York Times that Mueller's office made the concession in a letter to the president's legal team.
Mueller has reportedly not ruled out an in-person interview with Trump, though the letter did not ask for answers on obstruction of justice questions, leading team Trump to think that a sit-down with Mueller could be much less expansive in scope than they feared.
Trump's lawyers have dangled written answers as a possibility, and Mueller's team appeared receptive to it as an interim measure.
In appearances on television news programs, Giuliani has frequently discussed Mueller's pursuit of an interview with the president, seeming to wax and wane about the likelihood and desirability of such an interview.
Special counsel spokesman Peter Carr declined to comment Tuesday.
Defending champion Sloane Stephens knocked out by Anastasija Sevastova
Osaka will next face Lesia Tsurenko, the 36th-ranked Ukrainian who beat Czech teenager Marketa Vondrousova 6-7 (3-7), 7-5, 6-2. On Monday, Roger Federer and Maria Sharapova crashed out in back-to-back seismic shocks under the Arthur Ashe Stadium lights.
The Trump legal team's position has been that any sit-down interview has to be limited to matters before Trump became President, and for questions related to possible obstruction and other post-inauguration matters to be done in writing. In March, Mueller raised the prospect of subpoenaing Trump if he refused to be questioned voluntarily.
Echoing a previous argument, Giuliani said he will only allow Trump to submit to an interview if Mueller promises not to catch him in a "perjury trap". "Kavanaugh actually has a long résumé", including a decade as a federal judge and years as a lawyer in the George W. Bush White House", Trevor Noah said on Tuesday's Daily Show.
Trump claimed at the time that Mueller had multiple conflicts of interest, according to a January NYT report that relied on four unnamed sources.
Sources describe a lengthy response from Mueller with a strong emphasis on requesting written answers as it relates to the original mandate of the investigation - Russian meddling in the 2016 elections.
"How can you possibly have a man who is that", Trump said of Mueller.
The most important discrepancy relates to a meeting he had with Comey. You know, he's Comey's best friend, ' Trump said.