“I have heard on numerous occasions disturbing allegations by Rudy Giuliani about corruption in Ukraine and the many improprieties surrounding the firing of former Prosecutor General Viktor Shokin," Graham said in a statement. “Given the House of Representatives’ behavior, it is time for the Senate to inquire about corruption and other improprieties involving Ukraine."

It was unclear if Graham's invitation was for Giuliani to appear in public or behind closed doors. But a public hearing before lawmakers would give Democrats on the Senate Judiciary Committee — including presidential hopefuls Kamala Harris (D-Calif.), Cory Booker (D-N.J.) and Amy Klobuchar (D-Minn.) — an opportunity to interrogate him.

"I welcome the opportunity to question Rudy Giuliani under oath about his role in seeking the Ukrainian government’s assistance to investigate one of the president’s political rivals," said Sen. Dianne Feinstein, the top Democrat on the committee. "Democratic members have plenty of questions for Mr. Giuliani and this would give us an opportunity to help separate fact from fiction for the American people."

Later Tuesday, Giuliani signaled he was unlikely to appear before the committee. "Love Lindsey, but I am still a lawyer and I will have to deal with privilege," he told CNN.

Giuliani has spent months pressing Ukraine's leaders to open an investigation into baseless allegations that Biden engineered the firing of a Ukrainian prosecutor who was investigating an energy company with ties to his son, Hunter Biden. The prosecutor, in fact, had been viewed by the international community as an impediment to investigating corruption and Biden's efforts to have him removed were done in coordination with other Western governments.

Trump has echoed Giuliani's allegations about Biden and raised the issue in a July 25 phone call with Ukraine's new president Volodymyr Zelensky — a discussion that occurred amid Giuliani's pressure campaign and at a time the White House had quietly placed a hold on military assistance to Ukraine, which was eventually released after news of the hold became public.

Trump has tried to turn the discussion in Washington away from his efforts to pressure Zelensky and toward Biden.