Senate Minority Leader Charles Schumer Chuck SchumerPelosi slams Trump executive order on pre-existing conditions: It 'isn't worth the paper it's signed on' 3 reasons why Biden is misreading the politics of court packing Cruz blocks amended resolution honoring Ginsburg over language about her dying wish MORE (D-N.Y.) on Tuesday said that he supports a decision to invite Rudy Giuliani Rudy GiulianiThe Hill's Campaign Report: GOP set to ask SCOTUS to limit mail-in voting CIA found Putin 'probably directing' campaign against Biden: report Democrats fear Russia interference could spoil bid to retake Senate MORE, President Trump Donald John TrumpFederal prosecutor speaks out, says Barr 'has brought shame' on Justice Dept. Former Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. 1 Supreme Court pick MORE's personal lawyer, to testify before a key Senate panel, but said he must do so under oath.

"We welcome Mr. Giuliani testifying. Given the apparent depth of his involvement in the president’s effort to convince foreign governments to investigate a political rival, he must testify under oath," Schumer tweeted.

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Senate Judiciary Committee Chairman Lindsey Graham Lindsey Olin GrahamLincoln Project mocks Lindsey Graham's fundraising lag with Sarah McLachlan-themed video The Hill's Morning Report - Sponsored by Facebook - Republicans lawmakers rebuke Trump on election Trump dumbfounds GOP with latest unforced error MORE (R-S.C.) earlier on Tuesday announced that he would invite Giuliani to testify about Ukraine, marking a stark reversal from comments he made before Congress's two-week recess.

"It is time for the Senate to inquire about corruption and other improprieties involving Ukraine," he said on Tuesday.

Graham said his invitation was for Giuliani to "come before the Senate Judiciary Committee to inform the Committee of his concerns.”

Graham did not specify if the invitation was for Giuliani to meet privately with the panel and staff or to testify publicly.

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Under committee rules, as part of a deposition, an individual would be sworn in under oath. A spokesperson for Graham said the committee would also swear in a witness as part of a hearing.

Graham didn't specifically mention former Vice President Joe Biden Joe BidenFormer Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. 1 Supreme Court pick Bloomberg rolls out M ad buy to boost Biden in Florida MORE, one of the front-runners in the race to face Trump in the 2020 election, or his son, Hunter Biden, in his announcement. But a hearing would likely provide Trump's lawyer a high-profile perch to make his argument that Biden acted improperly in his dealings with Ukraine, though there's been no evidence of wrongdoing.

Democrats are also signaling that they would relish the opportunity to question Giuliani, who has emerged as a central part of the House Democrats' impeachment inquiry.

The Judiciary Committee includes three presidential contenders — Sens. Kamala Harris Kamala HarrisTexas Democratic official urges Biden to visit state: 'I thought he had his own plane' The Hill's Campaign Report: Biden on Trump: 'He'll leave' l GOP laywers brush off Trump's election remarks l Obama's endorsements A game theorist's advice to President Trump on filling the Supreme Court seat MORE (D-Calif.), Cory Booker Cory Anthony BookerThe movement to reform animal agriculture has reached a tipping point Watchdog confirms State Dept. canceled award for journalist who criticized Trump 3 reasons why Biden is misreading the politics of court packing MORE (D-N.J.) and Amy Klobuchar Amy Klobuchar3 reasons why Biden is misreading the politics of court packing Social media platforms put muscle into National Voter Registration Day Battle lines drawn on precedent in Supreme Court fight MORE (D-Minn.) — meaning 2020 politics will loom over any testimony.

Sen. Dianne Feinstein Dianne Emiel FeinsteinTrump plans to pick Amy Coney Barrett to replace Ginsburg on court Trump faces tricky choice on Supreme Court pick The Hill's Morning Report - Sponsored by Facebook - Trump previews SCOTUS nominee as 'totally brilliant' MORE (Calif.), the top Democrat on the committee, said she supported having Giuliani come before the panel.

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"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," she said.

House Democrats subpoenaed Giuliani late last month as part of an effort to force him to hand over documents, including text messages and phone records, regarding Hunter Biden and efforts by Giuliani or his associates to pressure current or former Ukrainian officials to investigate matters regarding the Bidens or any other American.

Giuliani told The Washington Post on Tuesday that he won't testify before the House panel.

—Updated at 3:07 p.m.