The letter is here, along with annotations by such anti-Trump New York Times reporters as Maggie Haberman.
Mr Trump had previously signalled that he would be willing to sit for an interview, but his legal team, including head lawyer Rudy Giuliani, have privately and publicly expressed concern that the president could risk charges of perjury.
The lawyers mounted a defense of Trump in what they outlined as 16 areas of Mueller's interest.
A court battle is likely if Mr Trump's team argues that the President can not be forced to answer questions or be charged with obstruction of justice.
Giuliani said Sunday that a decision about an interview would not be made until after Trump's summit with North Korean leader Kim Jong Un on June 12 in Singapore, and he cast doubt that it would occur at all. Negotiations between Trump's lawyers and the special counsel on a possible interview have continued since.
The possibility of a self-pardon appeared to be raised in a January 29 letter from Trump's lawyers to Mueller, published by the New York Times on Saturday, arguing that the president could not have obstructed the probe given the powers granted to him by the Constitution.
Not cool. The letter responded to Mueller's request that Trump agree to be questioned about allegations that he committed obstruction of justice.
In the letter penned by Trump lawyers John Dowd and Jay Sekulow, it was argued that President has the power to "order the termination of an investigation by the Justice Department or FBI at any time and for any reason". "If four of them are completely innocent and one of them is your assumption that it's a guilty motive, which (Trump) would deny, you can't possibly prosecute him".
And the letter also argues that the news media has not fully reported what is considered a damning statement by Trump to NBC anchor Lester Holt on why he fired Comey.
Trump tweeted 40 minutes before the New York Times broke the news, questioning whether Mueller's team or the Department of Justice was responsible for leaking the letter.
Twitter users immediately reacted to the news, also posting sarcastic remarks related to US President Donald Trump's comments on his Twitter page, in which he claimed that "more than 17 million dollars" had been spent on Mueller's inquiry and that the cost is "going up fast".
A spokesman for Mueller declined to comment to the Times.
They also spelled out in detail why the firing of James Comey, the Federal Bureau of Investigation director, and other key moments being scrutinised do not amount to obstruction of justice.
But it's not clear that Trump's lawyers understand those statutes, or the case, as well as they purport to.
"In light of these voluntary offerings, your office clearly lacks the requisite need to personally interview the president", the letter read.
The memo further argued that the president can terminate any investigation without obstructing justice.
It is unclear which letters the president was referring to.
Russian Federation has repeatedly denied allegations that it interfered in the 2016 USA presidential election, calling the accusations groundless. Multiple media outlets have additionally reported that Trump has erupted at his aides over Sessions on several occasions since a year ago, saying he needs an attorney general who will protect him.
In another episode Mueller has been probing, Trump's lawyers conceded for the first time that in July 2017, Trump "dictated" a statement to be released on behalf of his son Donald Trump Jr. about a meeting that the son had taken with a Russian lawyer during the campaign.