"Therefore I will offer to Mr. Giuliani the opportunity to come before the Senate Judiciary Committee to inform the committee of his concerns", said Graham, a close ally of President Trump, on Twitter.
In recent weeks, Giuliani has repeatedly alleged that Joe Biden, a leading Democratic presidential candidate, improperly pushed Ukraine to fire its top prosecutor in order to protect his son's business interests.
House Democrats are also at the start of an impeachment inquiry centered on Trump asking the Ukrainian government to work with Giuliani to look into the Bidens and allegations that the president tried to withhold aid to Ukraine in an effort to get Kiev to launch such a probe.
Then-Vice President Biden reportedly pressured the Ukrainian government to fire Shokin, who was investigating Hunter Biden, or else the USA government would withhold foreign aid to the eastern European nation.
Other Democrats on the Senate Judiciary Committee, though, have previously told ABC News they do not want to hear from Giuliani.
Giuliani said Volker and Gordon Sondland, the US ambassador to the European Union, came to him about the proposed statement on corruption and asked for his advice on what should be in it.
Giuliani said he is holding the Trump administration's line that the impeachment inquiry is "illegal". Cory Booker of Contemporary Jersey, Kamala Harris of California and Amy Klobuchar of Minnesota.
US envoy for Ukraine Volker RESIGNS amid Trump-Zelensky impeachment controversy Kurt Volker, US special envoy for Ukraine, has resigned from the position after he was named in a congressional subpoena as part of an impeachment probe into President Donald Trump over his dealings with Kiev.
Sondland routinely spoke to the president and had conversations with both Trump and Zelensky before and after the July 25 call, noting in a July 26 TV interview that he spoke to the president "a few minutes before" Trump made the July 25 phone call.
Giuliani, the former mayor of NY, has emerged as a central figure in the Democratic-led impeachment inquiry into Trump.
But at the very least, bringing Giuliani before the full committee would give multiple senators the chance to hold Giuliani's allegations up to the light, vetting the quality of the evidence he presents and identifying other witnesses whose testimony should be compelled.
A former New York City mayor, Giuliani said he would like to appear before the Senate panel but needed to determine whether maintaining confidentiality as Trump's lawyer would be an issue.