It is the first time the US leader has directly co-operated with the investigation he has long decried as a “witch-hunt”, with some fearful that his post-midterms sacking of attorney general Jeff Sessions would lead to an attempt to bring the inquiry to an end.
By submitting written answers to the special counsel, Mr Trump has – for now at least – avoided the prospect of a potentially risky face-to-face interview with prosecutors, which his lawyers have been keen to avoid.
Mr Trump had known for months that his legal team had been sent dozens of questions regarding the investigation, which aims to uncover whether his campaign team worked with the Kremlin to influence the election.
His lawyer, Jay Sekulow, said the questions answered in writing were solely related to the issue of Russia and did not take into account suggestions that the president has tried to obstruct the investigation – notably through his sacking of former FBI director James Comey.
Another of his lawyers, former mayor of New York Rudy Giuliani, said Mr Mueller had received “unprecedented co-operation” from the White House, including handing over around 1.4 million pages of documents.
But despite finally co-operating, the president still used a Fox News interview broadcast on Sunday to once again label the investigation as a “witch-hunt”.
He said: “I think we’ve wasted enough time on this witch-hunt and the answer is, probably, we’re finished.”
The investigation began in May 2017 and has already resulted in a number of indictments.
In August, former Trump campaign chairman Paul Manafort was found guilty of tax and bank fraud.
Within moments of that news breaking, Michael Cohen – a long-term lawyer of Mr Trump – pleaded guilty to violating campaign finance laws in connection to payments to two women, including former porn star Stormy Daniels.
The alleged aim of the payments was to influence the 2016 election.
Following his conviction, Mr Cohen was said to be “more than happy” to tell the special counsel everything he knows about the president.
The investigation is now being overseen by Matthew Whitaker, who became acting attorney general following the sacking of Mr Sessions earlier this month.
He could have the power to end the inquiry and is on the record in his view that it would be simple enough to do so.
In an interview with CNN last year, Mr Whitaker said: “I could see a scenario where Jeff Sessions is replaced with a recess appointment and that attorney general doesn’t fire Bob Mueller, but he just reduces his budget to so low that his investigation grinds to almost a halt.”
After his appointment, Senate Democrat leader Chuck Schumer said the president risked creating a “constitutional crisis” if he sought to impede the Mueller investigation.