Earlier this month, Trump alleged -- without providing evidence -- that Obama ordered his phones to be wiretapped at Trump Tower during the campaign. Graham and Sen. Sheldon Whitehouse, D-Rhode Island, both asked the Justice Department last week to provide any evidence that would support Trump's claims.
"We wrote a letter -- Sen. Whitehouse and myself -- wanting to know if there's evidence of a warrant issued by the Trump campaign," the South Carolina Republican told CNN's Chris Cuomo on "New Day." "He hasn't answered that letter or confirmed if there's a real investigation of the Trump campaign."
"He needs to answer the letter and give the nation some information about what's going on here," he added.
If the FBI does not provide the information, Graham said he and other senators may subpoena it.
"I want to get to the bottom of it. The FBI would know if a warrant was issued. They would know if a warrant was applied for. I want to answer that question," he said. "If they do not provide the answer to that letter we wrote in a bipartisan fashion, there will be a bipartisan subpoena following the FBI."
"Congress is going to flex its muscles," Graham added.
Graham later told CNN's Kate Bolduan that he has "no idea" if he will receive a response and he was not given any assurances that he would get an answer by Wednesday.
Graham -- who, along with Whitehouse, is leading one of the three congressional investigations into Russia's interference in the US election -- said the lack of an answer from the FBI is causing him to become suspicious.
"I've seen no evidence of collusion between the Trump campaign and Russia. I've seen no evidence of a wiretap being requested by the Department of Justice, a warrant," he said. "But the longer it takes to answer my letter, the more concerned and suspicious I'm beginning to be."
The senator added that if the FBI does not respond to lawmakers' requests for information, some may be willing to block more appointments to the Trump administration.
"If there's not a criminal investigation of the Trump ties to Russia, I think the country needs to know it. Congress needs to know it, and he has never promised to answer that question," Graham said. "Chairman (Chuck) Grassley said something very important yesterday, that he would be willing to block the deputy attorney general's nomination until Director Comey gave us the information we sought regarding warrants and investigations."
The White House is beginning to walk back
Trump's claim earlier this month that Obama had his "wires tapped" toward the end of the campaign. White House press secretary Sean Spicer said Monday the President wasn't referring to wiretapping when he tweeted about wiretapping.
"I think there's no question that the Obama administration, that there were actions about surveillance and other activities that occurred in the 2016 election," Spicer previously said. "The President used the word 'wiretaps' in quotes to mean, broadly, surveillance and other activities."