“Rudy Giuliani is a well-known man," said Graham. "He's a crime fighter. He's loyal to the president. He's a good lawyer.”
Giuliani is also under scrutiny by federal authorities. That means the Justice Department would be putting itself in the awkward position of appearing to work with someone it is actively investigating to gather potentially damaging information against one of the president's political rivals.
Justice Department spokeswoman Kerri Kupec declined to comment when asked about Graham's statements.
Federal prosecutors in New York are investigating Giuliani's business dealings, including whether he failed to register as a foreign agent, according to people familiar with the matter. They were not authorized to discuss the investigation publicly and spoke on the condition of anonymity.
Giuliani said Sunday that he has a document that relates to Biden's son Hunter, along with a memo allegedly from a Democratic Party official documenting communications with a reporter.
The Democratic-controlled House impeached Trump last year, alleging abuse of power and obstruction of Congress for pressing Ukraine to investigate the Bidens, while delaying delivery of nearly $400 million in congressionally approved security assistance for the Eastern European nation. Hunter Biden served on the board of a Ukrainian gas company while his father managed the U.S. government's Ukraine portfolio under President Barack Obama.
Trump was acquitted last week by the Republican-led Senate.
Graham said anything Giuliani has obtained from Ukraine needs to be given to the Justice Department or Congress' intelligence committees “because it could be Russian propaganda.”
Graham said he also spoke Sunday with Republican Sen. Richard Burr of North Carolina, chairman of the Senate Intelligence Committee. He said Burr also warned him to “take very cautiously” any information coming out of Ukraine. Graham said his message to Giuliani is: “Don't give it to me.”
A spokeswoman for Burr did not immediately return an email message seeking comment.
Graham was interviewed on CBS' “Face the Nation” and Giuliani appeared on Fox News Channel's “Sunday Morning Futures.”
Associated Press writer Michael Balsamo contributed to this report.