Speaking to reporters at the White House, Trump said he wrote the answers to the questions himself.
"My lawyers don't write answers. I write answers. I was asked a series of questions. I've answered them very easily," Trump said.
The Republican president did not specify when his legal team would submit his written responses. Peter Carr, a spokesman for Mueller, declined to comment.
"I'm sure they're tricked-up because, you know, they like to catch people," Trump said, referring to questions that he suggested could be designed to result in perjury charges. "You have to always be careful when you answer, with people that probably have bad intentions. Now, the questions were very routinely answered by me."
Mueller is investigating whether members of Trump's campaign conspired with Moscow in the 2016 election and whether Trump has unlawfully sought to obstruct the probe, which has cast a cloud over his presidency.
Trump and his lawyers had been in negotiations with Mueller's team for months over how the president would be questioned as part of the investigation.
Trump's team agreed to have the president submit written answers to questions relating to Russia's role in the election, but it is still possible Mueller may press Trump to answer questions on potential obstruction at some point. Another open question is whether Trump will eventually sit for an interview.
Tensions have risen over the probe since Trump last week ousted Jeff Sessions as US attorney general and named Matthew Whitaker as his replacement on an acting basis. That move gave Whitaker, a Trump loyalist, oversight of the Mueller probe.
Mueller already has brought charges against a series of former Trump aides, including his former campaign chairman and his former national security adviser, as well as a number of Russian individuals and entities.
A federal judge on Thursday refused to dismiss criminal charges against Concord Management and Consulting LLC, a Russia company accused by Mueller of funding a propaganda operation to sway the 2016 election in Trump's favor.
Trump on Friday again criticized the probe as a "witch hunt" and said there was no collusion. Russia has also denied any interference.
"There should have never been a so-called investigation, which, in theory, it's not an investigation of me. But ... as far as I'm concerned, I like to take everything personally because you do better that way," Trump added.
Democrats and a number of Republicans have raised concerns about Whitaker's appointment, with a bipartisan group of U.S. senators renewing a push for legislation to protect the special counsel. But Senate Majority Leader Mitch McConnell, a Republican, has opposed any such legislation.
Republican Senator Lindsey Graham, who has said he would seek to lead the Senate panel overseeing the Justice Department next year, on Thursday met with Whitaker and expressed confidence the Russia probe would continue.