On Friday, Fox also reported that then-Acting FBI Director Andrew McCabe and FBI lawyer Lisa Page were in the room when Rosenstein raised the subject.
Page had been half of the infamous texting romantic-affair couple who mused in 2016 about how to 'stop' Trump from becoming president.
An Obama-era Justice Department spokesman suggested Friday afternoon that McCabe leaked the story to the Times.
'Dangerous game Andy McCabe is playing right now,' Matthew Miller tweeted.
Ari Fleischer, who was White House press secretary during the George W. Bush administration, lashed out separately at McCabe.
'This story reads like Andy McCabe trying to burn down the house he once lived in,' he tweeted.
'Looks to me like McCabe is trying to get revenge on those he used to work with, after they challenged his honesty and fired him.'
McCabe is himself facing a federal probe over allegations that he misled investigators about the sources of press leaks; Attorney General Jeff Sessions terminated his employment this year, just days before he was scheduled to retire with a full pension.
In the Post's telling, McCabe had proposed opening an investigation into the president after the firing of FBI Director James Comey.
'What do you want to do, Andy, wire the president?' Rosenstein chided him, according to one source.