Trump fired Comey last week, suggesting his decision was motivated by the FBI's investigation into possible collusion between Russian Federation and his campaign.
Rosenstein appointed a special counsel to oversee the probe on Wednesday, in an escalation of the investigation that Trump called "the single greatest witch hunt of a politician in American history!"
As well as being accused of obstructing the investigation into Flynn and possible collusion with Russia, this week saw reports that Trump had disclosed highly classified information in a meeting last week with Russian Foreign Minister Sergey Lavrov and ambassador to the U.S. Sergey Kislyak. Trump has denied any collusion with the Russian government and no evidence has been publicly presented to show such collusion.
Trump repeated that he felt the probe was akin to a "witch hunt" during the Thursday press conference.
Trump also boasted of what he saw as the "tremendous success" of his presidency. Mr Trump said on national television last week that the Russian investigation, which he says is an attempt by Democrats to save face following an embarrassing electoral defeat, was a consideration in making the decision to can Mr Comey. "I actually thought when I made that decision, and I also got a very, very strong recommendation, as you know, from the Deputy Attorney General Rod Rosenstein".
The men were memorably cast - mostly via Comey's 2007 testimony - as honorable men standing against government overreach when they talked former President George W. Bush out of pushing ahead the renewal of a controversial surveillance program.More news: Skoda Karoq drip-feed continues with new close-up images
Lavrov said: "So, if you're talking about that, I see no secret here".
He later told senators of Mueller: "He's one of the finest people I've ever met".
One Republican official, who requested anonymity in order to speak freely, said after meeting Trump recently he did not think the president had a firm enough grasp on the nuances of the long-running Israeli-Palestinian conflict.
When Comey was initially called to Ashcroft's bedside, his first call was to Mueller who reportedly responded: "I'll be right there".
Some doubt whether Trump, a businessman-turned politician who never held elective office before becoming president in January, is ready for a smooth presidential debut overseas.