But Biden's further remarks about Graham drew a response on Twitter from Trump lawyer Rudy Giuliani, who wrote that Biden's comments amounted to "a threat to a U.S. Senator," and reminded the former federal prosecutor of "my old mafia cases."
During an interview with CNN's Don Lemon, Biden also criticized President Trump, claiming Graham cannot break with the president for fear of losing his Senate seat in 2020.
Lemon told Biden that Graham once called him the "nicest person" and recalled how the two worked together for many years.
After a pause, Biden turned to Lemon and said, "They're asking Lindsey Graham. They have him under their thumb right now. They know he knows if he comes out against Trump he’s got a real tough road for reelection."
On Thursday, Graham -- the chairman of the Senate Judiciary Committee -- sent a letter to Secretary of State Mike Pompeo requesting the release of any documents related to contacts between Joe Biden, Hunter Biden, and former Ukrainian President Petro Poroshenko.
Graham’s letter, which was released as the final day of public hearings in the House impeachment inquiry into Trump wrapped up, asked Pompeo to release a series of documents and transcripts of conversations in 2016 between Biden and Poroshenko in relation to an investigation into the Ukrainian natural gas company, Burisma Holdings.
Burisma – specifically, Hunter Biden’s role on the company’s board of directors – has become one of the main talking points from Trump and the Republicans amid the impeachment inquiry into the president.
In his interview with CNN, Biden said he is "quite frankly... angered" by Graham's move.
"He knows me. He knows my son. He knows there's nothing to this," the former Delaware senator said.
He claimed Trump is "essentially holding power over him [Graham] that even the Ukrainians wouldn't yield to."
"Lindsey is about to go down in a way that I think he's going to regret his whole life," he said, going on to address Graham directly: "Lindsey... I'm just embarrassed by what you're doing -- for you -- I mean, my Lord."
Giuliani, the president's lawyer, tweeted Friday that Biden's remarks constituted "a threat to a US Senator."
"This is getting to be more and more like my old mafia cases," Giuliani added. "They sure do sound like crooks."
Giuliani also wrote to Graham, telling him that "there is a very real set of crimes crying out to be investigated and, if necessary and appropriate, prosecuted."
In the letter, obtained by Fox News, Giuliani calls allegations against Biden "highly credible" and says that the conduct, if true, "is corruption at the highest levels of both the American and Ukrainian governments."
"I will supply your staff with the three names of the witnesses who need visas and ask that you cut through the State Department's obstruction, to make certain they can come here quickly and share their highly relevant evidence with us," he wrote. "I also will certainly be willing to give them the identity of the former United States Attorney, who is willing to share his surprise at the failure to follow-up in October 2018 to serious allegations of corruption."
"Someone should find out what happened," he said.
Earlier this year, former South Carolina Democratic Party Chairman Jaime Harrison announced he would challenge the three-term Graham in the 2020 election.
Prior to his election to the U.S. Senate, Graham served as a U.S. congressman from upstate South Carolina and played a key role as "manager" during the House of Representatives' Clinton impeachment investigation.
Fox News' Ed Henry, Adam Shaw and Andrew O'Reilly contributed to this report.