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 Russian Federation inquiry, the Washington Post reported.
The report came just one day after The New York Times said Mueller has a list of almost 50 questions for Trump aimed at trying to figure out if he obstructed justice.
"It'd be, max, two to three hours around a narrow set of questions", Giuliani, a former NY mayor who backed Trump in his presidential campaign said, according to the reporter's Twitter post.
Mr Trump could bring his own lawyers to a voluntary interview, but not to the grand jury, one possible point of leverage for Mr Mueller in the negotiations.
Trump has been open about his frustrations with the department for its handling of the special counsel's Russian Federation investigation. The occasion also was the first in which a sitting president testified under oath in a criminal investigation in which he was a potential target.
With those details in hand, Trump lawyer Jay Sekulow compiled a list of 49 questions that the team believed the president would be asked, according to three of the four people, who spoke on condition of anonymity because they were not authorized to talk publicly.
Additional reporting from Newsy affiliate CNN.
It's the latest shake-up for a legal team grappling with unresolved questions on how to protect the president from legal and political jeopardy in Mueller's Russian Federation probe, which is nearing its one-year anniversary.
The questions contained in a document posted online by the Times on Monday night reflected questions that defense lawyers anticipated rather than verbatim queries that Mueller's team provided, the person said.
Dowd's comments come more than a month after he resigned from the legal team, and they provide a new window into the nature of the Trump legal team's interactions with the special counsel, who the president has increasingly tried to undermine through public attacks.
Trump tweeted earlier that: "There was no Collusion (it is a Hoax) and there is no Obstruction of Justice (that is a setup & trap)".
Despite that, Trump wrongly tweeted on Tuesday that none of the leaked questions deal with any alleged "collusion" between his campaign and Russian Federation.
The questions are a reminder of just how aberrant this White House has been. "And if you don't have a crime, he doesn't get to ask superior executive officials a bunch of questions", said McCarthy, likening a Mueller subpoena to a subordinate military officer making demands of a superior officer.
The questions also involve key moments from the early months of the Trump administration, including his reaction to Attorney General Jeff Sessions' recusal from the Russian Federation investigation and Trump's firing of his national security adviser, Michael Flynn.
"What efforts were made to reach out to Mr Flynn about seeking immunity or possible pardon?" goes another. His view of Mueller soured further after raids last month that targeted one of his personal lawyers, Michael Cohen, in a separate investigation.