In one question obtained by The Times, Mr Mueller asks what Mr Trump knew about campaign staff - including his former campaign chairman Paul Manafort - reaching out to Moscow.
The questions also show Mueller wants to find out what Trump knew about links between members of his presidential campaign and Russia, including a June 2016 meeting at Trump Tower with a Russian lawyer offering damaging information on election rival Hillary Clinton, the Times said.
The New York Times obtained the list of questions, which are moistly related to a potential obstruction of justice case against Trump.
And they indicate that Mueller wants to know if Trump offered a pardon to his former national security advisor, Michael Flynn, who has already been charged by Mueller with lying and is believed to be cooperating with the investigation.
The brunt of what special counsel Robert Mueller plans to grill President Trump on - should he agree to an interview - is how he treated former Federal Bureau of Investigation director James Comey before and after his infamous firing.
Numerous questions obtained by The Times focus on the obstruction issue, including Mr Trump's angry criticism of Attorney-General Jeff Sessions' recusal from the Russian Federation investigation. Mueller planned to ask, according to the New York Times reporting.
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The questions also touch on the Russian meddling and whether the Trump campaign coordinated with the Kremlin in any way.
Mueller even plans on grilling Trump on specific tweets. The Mueller investigation then operated without any independent supervision from outside the agency, review by any elected officials or contemporaneous judicial review.
Speaking before a friendly audience, which often broke into applause, Comey spoke about his observations of Trump, Mueller's investigation and the impact Trump's presidency is having on the American political landscape.
Mueller also wants to find out Trump's thought process in deciding to fire Comey. "Who played a role?"
What did you know about phone calls that Mr. Flynn made with the Russian ambassador, Sergey I. Kislyak, in late December 2016?
Previously, Trump said he would be willing to sit down for an interview under oath with Mueller.