Although Mueller's team has indicated to Trump's lawyers that he's not considered a target, investigators remain interested in whether the president's actions constitute obstruction of justice and want to interview him about several episodes in office.
Many of the questions obtained by the Times center on the obstruction issue, including his reaction to Attorney General Jeff Sessions' recusal from the Russia investigation, a decision Trump has angrily criticized.
Trump lawyer Jay Sekulow declined to comment to The Associated Press on Monday night, as did White House lawyer Ty Cobb.
The questions also touch on Russian meddling and whether the Trump campaign coordinated in any way with the Kremlin. In one question obtained by the Times, Mueller asks what Trump knew about campaign staff, including his former campaign chairman Paul Manafort, reaching out to Moscow.
Mueller has brought several charges against Manafort, but none are for any crimes related to Russian election interference during the 2016 campaign. And he has denied having anything to do with such an effort.
The queries also touch on Trump's businesses and his discussions with his personal lawyer, Michael Cohen, about a possible Moscow real estate deal. Cohen's business dealings are part of a separate FBI investigation.
One question asks what discussions Trump may have had regarding "any meeting with Mr. Putin," referring to Russian President Vladimir Putin. Another question asks what the president may have known about a possible attempt by his son-in-law, Jared Kushner, to set up a back channel with Russia before Trump's inauguration.
Additional questions center on Michael Flynn, Trump's former national security adviser, who has pleaded guilty to lying to the FBI about his discussions on sanctions against Russia with Russian Ambassador Sergey Kislyak during the presidential transition. Flynn is now cooperating with Mueller's investigators.
"What did you know about phone calls that Mr. Flynn made with the Russian ambassador, Sergey I. Kislyak, in late December 2016?" reads one question. Another asks if there were any efforts to reach out to Flynn "about seeking immunity or possible pardon."
Flynn was fired Feb. 13, 2017, after White House officials said he had misled them about his Russian contacts during the transition period by saying that he had not discussed sanctions.
The following day, according to memos written by Comey, Trump cleared the Oval Office of other officials and encouraged Comey to drop the investigation into Flynn.