Mr Mueller suggested the move during talks with Mr Trump’s lawyers in March.
The threat to issue a subpoena, as it is known, was reportedly met with a sharp response from one of Mr Trump’s former lawyers.
It is believed to be the first time the special counsel has raised such a possibility.
Mr Trump’s lawyers insisted during the meeting that the president was under no obligation to face questions by federal investigators in relation to the Russia inquiry, the Washington Post reported.
However Mr Mueller’s team reportedly responded by suggesting they would issue a subpoena if Mr Trump declined. They agreed to provide the president’s lawyers with more specific information about the questions they wished to ask Mr Trump.
The president’s former lawyer, John Dowd, has also said that Mr Mueller mentioned the possibility of forcing Mr Trump to face questions.
Mr Dowd, who resigned about a week and a half after the meeting, said he told investigators that the probe was not “some game”, adding: “You are screwing with the work of the president of the United States.”
The list of possible questions has since been published in the New York Times, and it reportedly covers the president’s motivations in dismissing former FBI director James Comey last May and his campaign’s contacts with Russia.
Mr Trump has called the leak “disgraceful”, repeating his claim that the Russia inquiry is a “witch hunt”.
The US president himself has said he is willing to speak to Mr Mueller, but CNN reports that his enthusiasm has cooled after the offices of his personal attorney was raided.
The network cites sources close to Mr Trump as saying they are yet to make a recommendation about whether he agrees to an interview with Mr Mueller. Some advisers reportedly believe Mr Mueller would not go so far as to issue a subpoena.
If one was issued, Mr Trump’s lawyers could fight it in court, or he could refuse to answer questions by pleading the Fifth Amendment, a constitutional protection against potential self-incrimination.
Prominent US lawyer, Alan Dershowitz, who has expressed support for Mr Trump in his legal battles, advised against the US president speaking to the special counsel.
“The strategy is to throw him softballs so that he will go on and on with his answers,” he told the Washington Post.
“Instead of sharp questions designed to elicit yes or no, they make him feel very comfortable and let him ramble.”
The special counsel is looking into Russia’s efforts to interfere in the 2016 US election, whether there was any collusion between the Kremlin and Mr Trump’s election campaign and whether the president unlawfully tried to obstruct the inquiry.
Mr Mueller was appointed special counsel following Mr Trump’s firing of FBI director James Comey last May.
What does the special counsel do?
The president insists there was no collusion between his election campaign and the Russians.
US media reported that sources familiar with the case said Mr Mueller informed Mr Trump’s attorneys in March that he is a subject of the investigations, but not a criminal target.
It remains unclear when Mr Mueller will request a meeting with the president.