But Giuliani said he wants another investigation into the bureau's Russian Federation investigation, which was eventually taken over by Mueller when he was named special counsel.
"I think that we would prefer questions and written answers".
He went on to say that pardons weren't out of the picture, but would have to wait until the investigation was over. "Nobody is taking that away from him, and he can pardon in his judgment based on the Justice Department counsel's office", he continued.
But, he added, "When it's over, hey, he's the president of the United States".
Strzok was one of the lead investigators during the probe into Democratic presidential candidate Hillary Clinton's alleged use of private email server.
Giuliani also called on the Justice Department to investigate the Mueller probe, "not because of Mueller but because of its genesis", he said, referring to a report the inspector general put out last week on the FBI's Clinton email probe.
Giuliani, an outspoken attorney for the President, repeatedly suggested the probe had questionable origins and echoed some of the President's rhetoric about the investigation being unfair.
Many Republicans in Congress, privately and publicly, have urged the president not to consider pardons for anyone, including himself, at least until the investigation closes.
Giuliani's comments were the latest in a barrage of attacks on the probe by the president and his lawyers as Mueller appears to be nearing a conclusion to his investigation into possible collusion between Trump's campaign and Moscow.
Sen. Susan Collins of ME said on CBS' "Face the Nation" that she thought it was helpful Giuliani has recommended against pardons.
"Jake, I don't think that is the interpretation", Giuliani said. "Oh, I see, there was no Russian Collusion, so now they look for obstruction on the no Russian Collusion". "The phony Russian Collusion was a made up Hoax". Mr. Trump seized on the findings of the Justice Department Inspector General, claiming the report provides evidence of deep bias against him at the Federal Bureau of Investigation, and "exonerates" him from allegations of collusion with Moscow and obstruction of justice.
Signalling some openness to a Trump interview with Mueller's team, Giuliani outlined the possibility under narrow conditions and said he expected Trump to make a decision by July 4.
The inspector general's report said it would continue investigating FBI officials who received tickets to sporting games and other things of value from reporters.
"That's why you don't want to go under oath", Giuliani said.
The DOJ report pointed to the text messages Strzok sent to Lawyer Lisa Page, whom he was dating at the time, indicated "a biased state of mind".