The memo from William Barr was seen as a potential threat to the Mueller probe and possible insulation for Trump if Barr is confirmed by the Senate as the new attorney general, replacing Jeff Sessions.
Barr, a conservative Washington lawyer, said in the memo that Trump's May 2017 firing of FBI director James Comey after Comey rebuffed pressure to ease off the Russia collusion probe was legal under presidential powers and not obstruction, according to The Wall Street Journal.
In the memo, which was sent to the Justice Department in June, Barr said that Mueller's inquiry into obstruction was "fatally misconceived" and "grossly irresponsible," the Journal said.
The obstruction question being pursued by Mueller's team is viewed as one of the largest legal threats to Trump's presidency.
"This is ominous: Barr's unsolicited critique of Mueller's pursuit of Trump's obstruction of justice needs to be challenged in the Senate Judiciary Committee," said Laurence Tribe, a professor at Harvard Law School.
Threat to Mueller probe seen
Barr, who was attorney general in 1991-93 under president George H.W. Bush, sent the memo to the Justice Department as a "former official" offering his views.
His strong conclusion takes on greater weight now that Barr has been chosen to be attorney general, where he will oversee the Mueller probe and have the ability to curb its actions.
When Barr wrote it, Trump was already known to have soured on Sessions over the investigation, repeatedly attacking his attorney general for having recused himself from overseeing it.
Trump finally forced Sessions to resign in November.
Trump also knows Barr: in 2017 and 2018 the president had considered hiring him as his own lawyer against the investigation.
CNN reported that Barr discussed the June memo with Trump before he was officially nominated.
The issue could pose a challenge when Barr's nomination is weighed by the Senate Judiciary Committee in January, and lead to calls for Barr to recuse himself from the investigation as Sessions had done.
The Journal reported Barr also shared his memo with a top White House lawyer dealing with the Mueller investigation in June.
"I would think that might make his supervision of the Mueller investigation, uh, problematic," Benjamin Wittes, a senior fellow at the Brookings Institution and a Washington legal analyst, wrote on Twitter.
Justice Department spokeswoman Kerri Kupec said the memo was unsolicited and not based on any inside information from the Mueller probe.
She also said that Justice Department ethics officials have already advised that the memo "would present no conflict as to his duties as attorney general."
"Mr. Barr has stated that, if confirmed, he will make any decisions based on the actual facts and circumstances of any particular matter."