Asked whether it was the right time to have someone such as Rogers or Cornyn, Graham flatly said, "no".
Two White House officials said some of Trump's frustration centers on what he views as unfair coverage of his decisions and overly harsh criticism of press secretary Sean Spicer, as well as deputy press secretary Sarah Sanders, who led much of the response to Comey's firing. Minority Leader Chuck Schumer of NY said on NBC's "Meet the Press" on Sunday that the choice should be "certainly somebody not of a partisan background, certainly somebody of great experience and certainly somebody of courage".
This comes as Democratic members of congress, and even at least one Republican according to the New York Times, are increasingly calling for just that.
Trump told reporters travelling with him on Air Force One on Saturday that he may be able to make his decision on the Federal Bureau of Investigation head before he leaves in his first foreign trip later next week. The FBI has never had a female director.
But Rosenstein seems to be sticking to what he previously told the Senate Judiciary Committee in March, which is that he would only appoint a special prosecutor if 'it's appropriate based upon the policies and procedures of the Justice Department, ' according to HuffPost.
Justice Department officials began interviewing candidates on Saturday to replace fired FBI Director James Comey. Mark Warner of Virginia, the top Democrat on the Senate Intelligence Committee, said his panel or another committee would "absolutely" subpoena the tapes.
When pressed about whether Trump recorded his conversations with Comey or others, Spicer responded: "The president has nothing further to add on that".
Several people close to the president say his reliance on a small cadre of advisers as he mulled firing Comey reflects his broader distrust of many of his own staffers.
Since Comey was sacked on Tuesday, Democrats have said the firing could be linked to the FBI's investigation of Trump campaign officials' potential ties to Russian operatives. Changing rationales for the firing offered by White House aides added an element of chaos to the president's action.
Acting FBI Director Andrew McCabe, Republican Senator John Cornyn, New York Appeals Court Judge Michael Garcia and former Assistant Attorney General Alice Fisher will be interviewed on Saturday for the post, an administration official said.
The next director will immediately be confronted with oversight of an FBI investigation into possible coordination between Russian Federation and the Trump campaign, an inquiry the bureau's acting head, Andrew McCabe, has called "highly significant". He is one of at least five people expected to interview for with Attorney General Jeff Sessions and his deputy, Rod Rosenstein, on Saturday.
US District Judge Henry E Hudson, a Bush appointee who struck down the centrepiece of the Obama administration's health care law in 2010.
-Adam Lee, special agent in charge of the FBI's office in Richmond, Virginia.
President Donald Trump is considering almost a dozen candidates to succeed ousted FBI Director James Comey, choosing from a group that includes several lawmakers, attorneys and law enforcement officials.
Meanwhile, the FBI Agents Association urged Trump to pick former Michigan Rep. Mike Rogers, saying his diverse background makes him the best choice for the job. When Rosenstein was nominated as deputy attorney general, the Senate confirmed him by an amazingly bipartisan margin, 94-6.
Garcia, a former NY prosecutor, held high-level positions in the Commerce Department, the Justice Department and the Department of Homeland Security.
Sessions has faced questions over whether his involvement in Comey's firing violates his pledge to recuse himself from investigations into Russian interference in the election. The investigation is overseen by Dana Boente, the USA attorney in Alexandria, Va., who now also serves as the head of national security prosecutions at Justice Department headquarters.