Speaker Kevin McCarthy, under pressure from the hard right to take aggressive steps against the Biden administration, suggested on Tuesday that the House was moving toward opening an impeachment inquiry against President Biden, as Republicans search through bank records hunting for damaging information about the first family.

It was a striking escalation by Mr. McCarthy, who had previously resisted calls to impeach Mr. Biden, arguing that he had yet to see anything that would warrant such an extraordinary step. It comes as Republicans are stepping up their efforts to target Mr. Biden and his son Hunter, who is set on Wednesday to enter a plea in a tax case, while working to deflect attention from a potential third indictment of former President Donald J. Trump, their party’s presidential front-runner.

Just last month, Mr. McCarthy and other top Republicans in the House turned back an attempt from the hard right to move to quickly impeach Mr. Biden over his handling of the border, arguing that more investigation was needed before taking such a consequential step.

Republicans are deeply divided on an impeachment of Mr. Biden, which lacks the votes to pass the House. More mainstream Republicans — particularly those from politically competitive districts that Mr. Biden won — and those aligned with party leaders have urged caution.

Impeachment proceedings take up a tremendous amount of floor time, making little else possible in Congress while they take place. Some Republicans have also argued that the House must find actual corruption or wrongdoing before lawmakers consider impeachment.

Any impeachment proceeding also would be sure to fall flat in the Senate, which is controlled by Democrats.

But Monday night on Fox News, Mr. McCarthy sounded more serious about impeaching Mr. Biden because of evidence uncovered during the House Oversight and Accountability Committee’s investigation into the financial dealings of his son Hunter, which Republicans have sought to use to allege wrongdoing on the part of the president himself.

“We’ve only followed where the information has taken us,” Mr. McCarthy told Sean Hannity, the Fox News host. “This is rising to the level of impeachment inquiry, which provides Congress the strongest power to get the rest of the knowledge and information needed.”

On Tuesday at the Capitol, he clarified that he was not yet announcing an impeachment inquiry, but strongly suggested that things were heading in that direction.

“You’ve got to get to the bottom of the truth, and the only way Congress can do that is go to impeachment inquiry that gives Republicans and Democrats the ability to get all the information,” he said.

No proof has emerged to link President Biden to the crimes his son is accused of, but that has not stopped Republicans from insinuating at every turn that he is corrupt and has covered up wrongdoing. On Tuesday, Mr. McCarthy compared him to President Richard M. Nixon, whose elaborate cover-up in the Watergate scandal led to his resignation.

Hunter Biden is expected to enter into a plea deal for misdemeanor tax crimes on Wednesday, but House Republicans and conservative groups are seeking to intervene in the case, urging a judge to throw out the agreement he reached with prosecutors.

At the same time, Mr. Trump has said he expects to be charged in a third criminal case, perhaps the most significant one to date, by the grand jury investigating the Jan. 6, 2021, attack on the Capitol and his efforts to overturn the results of the 2020 election.

At the Capitol on Tuesday, Mr. McCarthy cited several areas where he believed Mr. Biden and his administration had acted improperly that warranted further investigation and could rise to the level of an impeachment inquiry.

Mr. McCarthy said Mr. Biden had made several false or misleading statements about his son’s business dealings. For instance, during the 2020 presidential debate, Mr. Biden claimed that no one in his family had received money from China, when, in fact, Hunter Biden and his business associates received millions through their dealings with a Chinese firm.

Mr. McCarthy also pointed to a document released last week by Senator Charles E. Grassley, Republican of Iowa, that contained unverified allegations from an anonymous source claiming that both Hunter Biden and his father had accepted millions in bribes.

The F.B.I. accused Mr. Grassley of risking “the safety of a confidential source” by releasing the document, and Lev Parnas, an associate of Mr. Trump’s personal lawyer Rudolph W. Giuliani, who was assigned to dig up dirt on the Bidens in Ukraine, said Mr. Grassley was spreading “conspiracy theories.”

Mr. McCarthy’s comments came the same day that Representative Marjorie Taylor Greene, a Georgia Republican who is among the most hard right members of Congress, took to the House floor to renew her call for impeaching the president.

Ms. Greene filed impeachment articles against Mr. Biden shortly after he took office in 2021, and she has been agitating for aggressive actions against the president for years. She has built her influence on Capitol Hill, and forged a tight alliance with Mr. McCarthy, who consults her often on politics and policy.

Recently, Ms. Greene has begun a push to expunge the two impeachments against Mr. Trump, a largely symbolic action that lacks full Republican support in the House.

“What I’m demanding is that the Republican-led House of Representatives move forward on an impeachment inquiry on Joe Biden, because this type of corruption should never be allowed to stand,” Ms. Greene said on the House floor. “We must expunge President Trump’s wrongful impeachments, and we must impeach Joe Biden.”

Ian Sams, a White House spokesman, denounced Republicans for suggesting they would impeach Mr. Biden.

“Instead of focusing on the real issues Americans want us to address like continuing to lower inflation or create jobs, this is what the @HouseGOP wants to prioritize,” Mr. Sams wrote on Twitter. “Their eagerness to go after @POTUS regardless of the truth is seemingly bottomless.”