(CNN) President Donald Trump may have committed crimes over a scheme to pay off women alleging extramarital affairs -- but Senate Republicans say they have no reason to look into it.

GOP leaders and key committee chairmen are making clear that they believe there is no reason to probe whether the President broke the law in engaging in a scheme to hide payments made to two women to keep their stories quiet in the days running up to the 2016 elections.

Sen. Ron Johnson, who chairs the Homeland Security and Governmental Affairs Committee and investigated Hillary Clinton's email controversy in the last Congress, says he wants to wait until special counsel Robert Mueller finishes his investigation first.

Told that the investigation into the hush-money scheme was being led by the US Attorney's Office for the Southern District of New York, Johnson said: "We'll let the justice system work its way. ... I want to see the definitive information as opposed to show-trial type testimony at congressional hearings."

Trump met Tuesday with Senate Judiciary Chairman Lindsey Graham of South Carolina and Georgia Rep. Doug Collins, the top Republican on the House Judiciary Committee, to discuss border security and immigration ahead of a Wednesday Senate hearing. But Trump also discussed "harassment stuff" with the lawmakers, Graham told CNN.

"He just believes they are out to take a wrecking ball to his life," Graham said of Dems. "They'll go nuts."

Graham said he told Trump to "listen to his lawyers" about whether to comply with the Democratic requests. "You just have to put your head down, fight back and govern the country — that's what (Bill) Clinton did," Graham said, adding that Trump should challenge Democrats to "fix problems."

Graham said concerns about the Democratic probes comprised about "10%" of the meeting. Asked whether Trump suggested he should fight back, Graham said there was little that he could do, but he planned to use the Senate Judiciary Committee to investigate "abuse" under the Foreign Intelligence Surveillance Act, which would continue the GOP-led House probes in the last Congress into accusations of FBI and Justice Department misconduct.

The decision to ignore allegations that Trump may have broken the law -- and wrote a check reimbursing his former personal attorney Michael Cohen for the payments while serving as President -- highlights the stark contrast between the priorities of the Democratic-run House and Republican-run Senate over the next two years. Democrats say the GOP is ignoring its constitutional duties to oversee the Executive Branch -- and with several powerful committees ramping up oversight over all aspects of the President's political life, business career, campaign, transition and inauguration.

House Intelligence Chairman Adam Schiff announced Tuesday several new committee hires, including Daniel Goldman as a senior adviser and director of investigations. The hiring is notable because Goldman worked from 2010-2017 for the Southern District of New York — which charged Cohen over the hush-money payments — where Goldman was deputy chief of the organized crime unit.

Republicans say Democrats are overreaching and setting the stage for impeachment.

"I think what the Democrats are worried about now is the hysteria over the alleged collusion, that they're worried the Mueller report will come up with basically nothing against the President," said Texas Sen. John Cornyn, a member of the Senate Judiciary Committee and adviser to the GOP leadership. "That's why I think their full-frontal assault, which as I said I think is a prelude to impeachment and will make sure that we can't get anything else done."

Sen. John Kennedy, a Louisiana Republican on the Judiciary Committee, said the hush-money payments that Trump's former lawyer Michael Cohen said the President repaid while in office wouldn't be on his "top five" list of issues to investigate, because the Mueller and the federal prosecutors in New York are already probing it.

"I think there are some House members that decided they want to start the election early, and they're going to do everything they can to tear down the President," Kennedy said. "I'm not saying that the Republicans wouldn't do it if the shoe were on the other foot — that's part of the problem with this place."

In a statement Monday, White House press secretary Sarah Sanders blasted the new House investigation as "a disgraceful and abusive investigation into tired, false allegations already investigated by the Special Counsel and committees in both Chambers of Congress."

Trump weighed in on the Democratic probes later Tuesday, telling reporters, "The witch hunt continues," and adding that "basically" Democrats have "started the campaign."

Trump also appeared to suggest that his predecessor hadn't complied with congressional investigations into the Obama administration.

"President Obama, from what they tell me, was under a similar kind of a thing. They didn't give one letter. They didn't do anything. They didn't give one letter of the requests. Many requests were made. They didn't give a letter."

House Democrats are rapidly ramping up their investigations into all elements of Trump's world, from his businesses to his administration. After Cohen's blockbuster testimony last week — where he displayed a check Trump paid him in 2017 as partial reimbursement for the October 2016 payment to Stormy Daniels — three House committee said they wanted documents and testimony from Trump Organization chief financial officer Allen Weisselberg.

Weisselberg was one of 81 people and entities that received letters from House Judiciary Chairman Jerry Nadler on Monday asking for documents on a wide-range of issues, marking the start of a sprawling investigation into possible corruption, obstruction of justice and abuse of power.

One Senate committee continues to probe possible collusion between the Trump campaign and Russia in 2016: The Senate Intelligence Committee, which is two years into its investigation into Russia's 2016 election meddling. The panel also interviewed Cohen last week behind closed doors, the day before his high-profile public testimony, but lawmakers say the President's role in hush-money payments is beyond the scope of the committee's probe.

In 2017, the GOP-led Senate Judiciary Committee did launch an investigation into the circumstances surrounding the June 2016 Trump Tower meeting, where Donald Trump Jr. was promised "dirt" on Clinton's campaign from a Russian lawyer. That probe wrapped up last year amid infighting between Democrats and Republicans over witnesses, with Democrats demanding Trump Jr. and others return for public testimony.

"The Senate Judiciary Committee has a responsibility to be more aggressive in continuing its investigation which unfortunately we short-circuited," said Sen. Richard Blumenthal, a Connecticut Democrat who sits on the panel. "We need to bring back Donald Trump Jr. and anyone else who has known about those hush money payments."

Republican Sen. Chuck Grassley, a senior member of Senate Judiciary who chaired the panel in the last Congress, wouldn't say if he had concerns about Trump's involvement in a hush-money scheme.

"You always ask me questions that I haven't studied," Grassley said Tuesday. "So I can't say."

A fellow Iowa Republican on the Judiciary panel, Sen. Joni Ernst, also suggested she wasn't fully aware of the hush money when asked if the committee should look into the scandal.

"Possibly at some point -- I don't know enough about it," Ernst said.

On the House side, Republicans can do little while in the minority to stop their Democratic counterparts. Instead, they've vocally criticized their Democratic colleagues for trying to tear down the President with their investigations. Trump has also frequently slammed them on Twitter as "presidential harassment."

After attacking Cohen's credibility at his hearing last week, Republicans downplayed the significance of the hush-money allegations.

"I think it's news we knew about," Ohio Rep. Jim Jordan, the top Republican on the House Oversight panel, said after the hearing concluded.

House Minority Leader Kevin McCarthy accused Nadler, whose committee would lead impeachment proceedings, of deciding to impeach Trump "the day the President won the election."

Asked if he was concerned about the President directing the hush-money payments, McCarthy instead put the blame on Cohen.

"You know what concerns me? If you hire an attorney -- if I hire an attorney to make sure I carry out the law, the attorney has a responsibility to tell me what's right and wrong in the process," he told ABC News. "If it's a finance campaign, those are fines. Those aren't impeachable in the process."

Some of the President's allies are also not eager to engage in the controversy.

Asked Tuesday if he had any concerns about Trump paying off women and allegedly breaking the law, Sen. Ted Cruz of Texas ignored the question and kept walking.

This story has been updated with additional developments Tuesday.