Graham, in a series of television interviews, said he was prepared to issue subpoenas for information on whether the Federal Bureau of Investigation received a warrant to tap Trump's phones during the 2016 presidential campaign, as the president alleged on Twitter, without offering proof.
Additionally, Charles Grassley, the chairman of the Senate Judiciary Committee, has threatened to hold up Trump's nominee for deputy attorney general, Rod J. Rosenstein, until Comey briefs the panel about alleged Russian ties to Trump.
"Congress is going to flex its muscles," Graham said on CNN.
Graham said he and Democratic Senator Sheldon Whitehouse, both Judiciary Committee members, asked Comey two weeks ago to provide the information by Wednesday.
U.S. intelligence agencies have concluded that Russia conducted cyber attacks on Democrats in an effort to influence the 2016 U.S. election on Trump's behalf. Russia has denied this.
At the same time, Trump has been dogged by allegations that his advisers or associates had ties to Russian officials. Trump fired his national security adviser, Michael Flynn, last month after he failed to disclose contacts with Russia's ambassador before Trump took office on Jan. 20.
Several congressional committees are looking into possible Russian interference in the Nov. 8 election, as well as Trump's wiretapping claim. Obama administration representatives have denied the claim.
Congressional Democrats want a special prosecutor or nonpartisan select committee to investigate any Russian ties. But Trump's fellow Republicans, who control majorities in both the Senate and House of Representatives, say the probes by congressional committees are sufficient.
"The Congress needs to know what it's doing," Graham said on MSNBC. "If you're going to have multiple congressional investigations and there is a criminal investigation, we're going to run into it.
"I think the entire country needs to know if there's something there."
Democratic Senator Tim Kaine, appearing on CNN, said he believed Comey will say, "Yes, the FBI is looking at it. That's what I expect to hear."
"We have to understand everything about the ties between Russia, the Trump campaign, the Trump transition, the Trump administration," said Kaine, a member of the Foreign Relations Committee. "Especially the degree to which Russia tried to invade the American election, because we have to protect future elections."
James Clapper, the director of national intelligence under President Barack Obama, has told NBC he knew of no warrant to wiretap Trump Tower. Graham said he had no reason to believe a judge ever issued a warrant, which would have reflected there was probable cause that the Trump campaign was colluding with the Russians.
If it is not true, he said, the FBI should confirm that.
"The longer it takes to answer that question the more suspicious I get," Graham said on MSNBC.