Senate Republicans are scrambling to contain the political fallout from reports that President Trump pressured a foreign leader to investigate his leading Democratic rival, former Vice President Joe Biden.

Several Republican lawmakers have called on Trump to reveal more details from his conversation with Ukrainian President Volodymyr Zelensky, in which the president on Sunday acknowledged discussing Biden and his possible links to corruption in Ukraine. This effort comes as some Democrats in the House are ramping up their calls for a vote on an impeachment inquiry.

Senate Majority Leader Mitch McConnell (R-Ky.) announced on the Senate floor Monday afternoon that Senate Intelligence Committee Chairman Richard Burr (R-N.C.) is trying to bring the Trump-appointed intelligence community's inspector general who received a complaint from a whistleblower before his panel to investigate the matter.

But McConnell cautioned his colleagues to handle the issue with bipartisan cooperation and refrain from some of the political fireworks that have erupted in the House over the complaint.

"I believe it's extremely important that their work be handled in a secure setting with adequate protections in a bipartisan fashion, and based on facts rather than leaks to the press," he said.

Senate Republican Whip John Thune (S.D.) told reporters Monday that Congress needs to be briefed on the whistleblower complaint.

"I would hope that whatever information is available that is in the possession of the inspector general of the DNI [director of national intelligence] that we would get access to that," he said.

"If there is some transcript or something available from that conversation, hopefully eventually we'll see it and everybody will be able to know what was said," he added. "The president has described it as a very appropriate conversation, but I think we'll see what the facts say."

A congressional official said intelligence community Inspector General Michael Atkinson and acting Director of National Intelligence Joseph Maguire are both expected to testify before the full committee later this week. Maguire is scheduled to appear before the House Intelligence Committee on Thursday.

Sen. Joni Ernst (R-Iowa), the vice chairwoman of the Senate Republican Conference who faces a potentially competitive election next year, said the appropriate path is to task the Intelligence Committee with getting to the bottom of the controversy.

"I just think all this needs to go to the Intel Committee and needs to be sorted out," she said.

Sen. Susan Collins (R-Maine), a member of the Intelligence Committee, said it's "important and appropriate" that the intelligence community's inspector general brief the Senate and House committees.

"The law is clear that if the inspector general of the intelligence community receives a whistleblower complaint that the IG [inspector general] deems is urgent that that is to be reported to the leaders of the Intelligence committees," she said.

GOP senators including Sen. Mitt Romney (Utah) and Pat Toomey (Pa.) are warning that Trump would have acted improperly if he in fact pressured Zelensky to investigate Biden, who is leading the Democratic field of presidential contenders.

"If the President asked or pressured Ukraine's president to investigate his political rival, either directly or through his personal attorney, it would be troubling in the extreme. Critical for the facts to come out," Romney, a Trump critic, tweeted Sunday. Romney said Monday that the transcript of Trump's call with Zelensky should be released.

Toomey on Sunday said on NBC's "Meet the Press" that it is "not appropriate" for a candidate to ask a foreign leader for assistance.

Rep. Adam Kinzinger (Ill.), a Republican member of the House Foreign Affairs Committee, said Monday that it would be "highly inappropriate" if Trump directed a foreign leader to investigate a rival politician.

"It's absolutely highly inappropriate what President Trump did, and I don't care about the Biden arguments, I, you know, that's important, and we need to get to the bottom of it," Kinzinger said on the "Guy Benson Show."

Senate Democratic Leader Charles Schumer (N.Y.) on Monday ratcheted up the pressure on more Republicans to call for additional details about what Trump might have offered the Ukrainian president in return for probing Biden's son, Hunter Biden, who served as a paid board member for a Ukrainian energy company.

Trump on Monday denied that he threatened to hold back foreign aid to pressure Ukraine.

"I did not make a statement that 'you have to do this or I'm not going to give you aid,' " the president told reporters on the sidelines of the United Nations General Assembly.

McConnell later said on the floor that it is "regrettable" that Schumer had "chosen to politicize the issue."

A Republican House member who requested anonymity told The Hill that Trump should release the transcript of his call to put an end to speculation.

"I believe reviewing the transcript is necessary and appropriate to put an end to the rumormongering," the source said.

Schumer argued in a letter to McConnell that Republicans should demand the release of a complaint made by an unnamed whistleblower to the intelligence community's inspector general, Michael Atkinson, claiming that Trump made an inappropriate pledge to Ukrainian officials.

Schumer accused Senate Republicans of remaining "silent and submissive" and "shying away from this institution's constitutional obligation to conduct oversight."

"The Republican Senate's 'see no evil, hear no evil' attitude toward such a serious national security concern is unacceptable and must change. I call on you and the chairs of the relevant Senate committees to fulfill your constitutional duty," he wrote.

Maguire has so far refused to turn the document over to Congress, which Democrats say is a violation of law. In a letter to House Intelligence Committee Chairman Adam Schiff (D-Calif.), Maguire last week stated the complaint was not an "urgent concern" because it applied to stakeholders outside the intelligence community and did not relate to intelligence activity.

Schumer called on Senate Republicans Monday to hold hearings to find out what prompted the whistleblower complaint; issue a subpoena to compel the release of the complaint to Congress; request the White House to release a transcript of Trump's call with Zelensky; identify administration officials who delayed aid to Ukraine; and ask for a legal opinion from the Justice Department on sharing the whistleblower complaint with Congress.

Juliegrace Brufke and Jordain Carney contributed.