DiGenova, who says he has spoken to the president in the past 24 hours, is one of a number of trusted voices in the president's orbit urging him to remove Rosenstein in wake of his decision to authorize a raid on the president's personal attorney, Michael Cohen.
The president is taking the advice seriously and is openly considering the move, sources told ABC News.
"Rosenstein authorized what I consider to be an unconstitutional search of Mr. Cohen's office in New York," diGenova told ABC, calling the deputy attorney general, whom Trump nominated for the job, "disloyal" to the president.
The raid incensed the president, who declared the investigation a "disgrace" and a "total witch hunt.”
The president was so aggravated by the news that he has become “less inclined” to sit down for an interview with Mueller, sources close to Trump and his legal team told ABC News, something he had said he would "100 percent" do.
It would be a bad idea for Trump to sit down for an interview with Mueller, diGenova said, describing it as a "perjury trap."
"I have not discussed that with the president but I would certainly recommend that he not do that," diGenova said of any interview with Mueller. "It would serve no purpose at this point.
"You do not allow your client to be interviewed by a prosecutor who is acting in bad faith,” diGenova added. "Mr. Mueller is acting in bad faith. This is a perjury trap."
DiGenova accused Wray of disappearing after reports surfaced that the FBI missed tips that could have prevented the Parkland, Florida, school shooting in February.
The president's next move is anyone's best guess. He has said publicly on more than one occasion that he'd sit down with Mueller, but earlier this week mused aloud about firing him instead.
"Why don't I just fire Mueller? Well, I think it's a disgrace what's going on. We'll see what happens," Trump said, speaking in the Cabinet room Monday after the Cohen raid was first reported. "And many people have said, you should fire him."
But nearly all the president's confidants and his allies in Congress have warned against such a move for fear that it could create a constitutional crisis and calls for impeachment.
Meanwhile, a bipartisan group of senators is hoping to introduce a bill in the coming weeks that would protect the special counsel from being fired by the president.
DiGenova also said Trump will be adding a new high-profile attorney to replace Trump’s lead personal attorney handling the Russia probe, John Dowd, who resigned two weeks ago.
DiGenova was briefly considered for that role but, ultimately, could not take the job because of conflicts of interest, he said.