(CNN) Democrats across Capitol Hill are promising to sharply scrutinize the circumstances surrounding President Donald Trump's ex-attorney Michael Cohen pleading guilty Thursday to making false statements about the Russia investigation.

Republicans, for their part, significantly downplayed the news about Cohen, either saying they were unaware of the details and would wait for special counsel Robert Mueller to issue his report before weighing in -- or saying it has little to do with Trump.

In the House of Representatives, where Democrats will retake the majority early next year, prominent lawmakers from the party are vowing to use their newfound power to deeply investigate the issue.

The incoming House Intelligence Committee chairman said that his panel will try to bring in Cohen and investigate whether there was any money laundering by the Russians through the Trump Organization.

Rep. Adam Schiff, the top Democrat on Intelligence, said he would be "looking into this issue of whether the Russians possess financial leverage over the President of the United States."

"If Mr. Cohen misled the Congress about the President's business dealings in Russia deep into the campaign, it also means that the President misled the country about his business dealings, and that the Russians were apparently attempting to gain financial leverage over the potential President of the United States," Schiff told CNN's Christiane Amanpour.

He added later, "So, clearly we have a lot more work to be done. And just as clearly the President has misled the country about his financial dealings with the Russians."

The likely incoming chairman of the House Judiciary Committee, Rep. Jerry Nadler, told CNN that the Cohen plea raises serious questions about what the Russian government has over Trump.

"He lied to Congress apparently about dealings between Trump and Russia, that leads me to suspect that there are more dealings that the President wanted hidden," said Nadler, a Democrat from New York. "And this raises all kinds of questions with respect to the question of whether the Russian government has any kind of hold over the President because of his real estate dealings, and whether the President knew about the obvious collusion between this campaign and Russians."

Asked if this raises impeachment possibility, Nadler said: "That obviously depends on what we find."

In the Senate, Democrats are also demanding answers and raising questions. Sen. Mark Warner, the top Democrat on the Senate Intelligence Committee, said Thursday's news is a "trend" emerging among so many of the President's closest allies.

"This is one more example of one of the President's closest allies lying about their ties to Russia and Russians," Warner said. "In this first blush looking, just remarkable that you have the President's personal lawyer still dealing on a Trump Tower project through the whole campaign, through 2017."

A Senate Intelligence Committee source said that in light of Thursday's developments, staffers were going through Cohen's testimony to the committee to determine whether he may have made any other false statements that warrant further scrutiny.

Warner says the committee still has a number of witnesses to see and they are working through how to get them to come back and that Cohen is one of the witnesses they always wanted to have back.

"As we continue to see this rollout of close Trump allies plead guilty, almost always to trying to hide their ties with Russia and with Russians and the Russians' attempt to infiltrate the Trump campaign -- I think there is more story to be told," Warner said.

"What are they covering up for?" Warner added. "Why are all his closest associates being found guilty about lying about their ties to Russia?"

Cohen, Trump's former lawyer, pleaded guilty Thursday to making false statements to Congress about the Russia investigation in a charge brought by Mueller. Cohen had previously said talks about the Moscow project had ended in January 2016 just prior to the Iowa caucuses.

"He's a weak person," Trump said before departing for Buenos Aires, Argentina.

Republicans distance Trump from Cohen

Even though Cohen said that discussions over the Trump Tower Moscow project extended into the heat of the campaign season, June 2016, and he had spoken more extensively to Trump about it than he had previously acknowledged to Congress, a number of Republicans sought to put distance between Trump and Cohen.

"All of that is just speculation," said Senate Majority Whip John Cornyn of Texas when asked about Trump's knowledge of the project during the campaign season.

Sen. Lindsey Graham, a Republican from South Carolina, referred to it as "a process crime," saying, "I don't know how this has got anything to do with colluding with the Russians in terms of the campaign."

"No," Sen. Thom Tillis of North Carolina said when asked if he shared any concerns that Russians may have financial leverage over Trump, as Democrats are alleging. "We're in the perpetual political season right now."

"I think this is another log in the fire of speculation," said Sen. John Kennedy of Louisiana, a member of the Senate Judiciary Committee, when asked about any concerns over Trump's business dealings with Russia during the campaign. "Eventually sooner or later, Mr. Mueller will issue a report."

Sen. Marco Rubio, a Florida Republican who's a member of the Senate Intelligence Committee, said this when asked about Trump's knowledge of the project: "I'm going to focus on our work in the Intelligence Committee and we'll let the report speak for itself when it's ready to come out, and my sense is that, if new information emerges, it'll extend the time frame it'll take to finish our work, but I'd rather not comment on it till we're done."

Sen. Ron Johnson of Wisconsin, chairman of the Homeland Security and Governmental Affairs Committee, who has attacked the FBI's handling of the Russia investigation, pleaded ignorance of the Cohen news.

"You're too young to remember 'Hogan's Heroes,'" Johnson said. "Sergeant Schultz says, 'I know nothing. I know nothing.' I really have no comment on it."

This story has been updated to include additional comment from Senate Republicans.