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.
Trump lawyer Jay Sekulow declined to comment to AP, as did White House lawyer Ty Cobb.
The questions also touch on the Russian meddling and whether the Trump campaign coordinated with the Kremlin in any way.
In one question obtained by the New York Times, Mueller asks what Trump knew about campaign staff, including his former campaign chairman Paul Manafort, reaching out to Moscow.
Manafort has not been charged with any crimes related to Russian election interference during the 2016 campaign. And he has denied having anything to do with such an effort.
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.
-AP