Graham said on NBC's "Meet the Press"
that Trump should use this moment to "reset" the situation surrounding the Department of Justice's investigation into alleged collusion between the Trump campaign and Russia.
"The President has a chance to clean up the mess that he mostly created," Graham said. "He really, I think, did his staff a disservice by changing the explanation."
After the news broke of Comey's ouster, Graham said he supported the move. But in the days afterward, the White House struggled to maintain a consistent explanation for the firing, with Trump contradicting his communications team
and saying it was "not possible" for them to be accurate all the time.
Trump also tweeted a warning
to Comey on Friday, saying he "better hope that there are no 'tapes' of our conversations before he starts leaking to the press."
In his news briefing later that day, White House press secretary Sean Spicer repeatedly declined to explain what the tweet meant or address questions about whether there were recording devices in the Oval Office.
Graham said if any tapes existed, Trump needed to turn them over, and added that the situation demanded the senator take further steps.
"I think it requires somebody like me, a Republican, to call Comey before the (Senate) Judiciary Committee," the South Carolina senator said.
Graham also told Trump to "back off" from commenting or tweeting about the investigation, adding that he did not believe Trump was personally "a target or subject of any investigation regarding collusion with the Russians."
In addition to any potential tapes, Graham said Trump should turn over his tax returns.
"I don't have a reason to subpoena them," Graham said. "If I get that reason, I'll do it. But he should turn over his tax returns. He should do that now."
Graham is leading an investigation into Russian interference in the US election, which included a hearing
featuring former acting Attorney General Sally Yates and former Director of National Intelligence James Clapper last week; and the senator has received a letter
from Trump's tax attorneys asserting that the past 10 years of his federal tax returns "with a few exceptions" don't show any income from Russian sources. Trump has refused to make his tax returns public, breaking with decades of presidential tradition.
Graham also said Trump should forgo nominating a politician to be the new FBI director and instead promote someone from within the bureau.
"It's now time to pick somebody that comes from within the ranks or is of such reputation that has no political background at all," Graham said.
Referring to reports that Trump is considering Sen. John Cornyn, the No. 2 Republican in the Senate, as well as several other candidates for the spot, Graham said the Texas senator, who is a former judge and attorney general of the state, was not the right choice.
"John Cornyn is a wonderful man," Graham said. "Under normal circumstances, (he) would be a superb choice to be FBI director. But these are not normal circumstances."