House Speaker Kevin McCarthy's recent comments about opening an impeachment inquiry into President Biden were greeted with skepticism from Senate Republicans, including one who warned against following the lead of Democrats who "cheapened" the process under former President Donald Trump.

"You'd have to have the argument," Senate Minority Whip John Thune, R-S.D., told reporters Tuesday. "It's a high threshold. I'd assume they'd have to have evidence and some process where they would at least get that evidence if they don't have it."

"Clearly the statements they're making would lead me to believe they have evidence. Or they think they have evidence that could reach that threshold of high crimes and misdemeanors," Thune added. "I'll say what I said before, and that is I think the best way to change the presidency is win the election… I think it's in our best interest to be making an argument for why we need to have the majority in the House, the Senate and the White House come Jan. 2025."

In a Monday night in an interview on "Hannity," McCarthy escalated his talk about potentially impeaching Biden by saying "this is rising to the level of impeachment inquiry." The speaker said evidence uncovered by House committees' investigations into the president may necessitate a full inquiry to gather more evidence of alleged corruption by the Biden family.

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House Speaker Kevin McCarthy

Speaker of the House Kevin McCarthy recently stepped up his talk about possibly impeaching President Biden. (Anna Moneymaker/Getty Images)

"You've got IRS whistleblowers saying something when it comes to government treating the Bidens different. You've got an informant claiming that the Biden family had been bribed. Should you ignore that or should investigate that?" McCarthy told reporters Tuesday. "The only way you can investigate that is through an impeachment inquiry. So the committee would have the power to get all the documents that they would need."

However, Senate Republicans were not excited about the prospect of a third presidential impeachment in four years.

"It's getting to be a habit around here, isn't it?" Sen. John Cornyn, R-Texas, said of impeachments.

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"No, it's not," Cornyn added when asked if that was a good thing. "Unfortunately what goes around, comes around. But obviously the stuff that the House is revealing about the Biden family business is very disturbing. But obviously the Senate doesn't have any role in that."

John Cornyn speaks to reporters

Sen. John Cornyn, R-Texas, said that impeachment is "getting to be a habit" in Congress. (Brandon Bell/Getty Images)

"I'll wait to see what evidence they present," Sen. Thom Tillis, R-N.C., said. "But we've got to do the homework. They cheapened the process the last two impeachments, and we don't want to repeat that mistake."

Sen. John Kennedy, R-La., said there are serious lingering questions about the Biden family, including what the president knew about his son Hunter's overseas business dealings and if Biden got any money from those deals. However, Kennedy said mere political disagreements should not be grounds for an impeachment.

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"No one should be impeached, certainly not a president, unless there is substantial evidence that the president has committed a high crime or misdemeanor," Kennedy said. "I'm not going to support — and I'm not suggesting this is what Kevin has suggested — but I'm not gonna support impeaching somebody just because I don't like their politics."

It is up to the House to impeach a president, but removal from office only happens if two-thirds of the Senate votes to convict the commander in chief.

President Joe Biden

Republicans in the House are moving closer to starting an impeachment inquiry into President Biden. (Ting Shen/Bloomberg via Getty Images)

An impeachment would be very difficult to get through the House with McCarthy's narrow GOP majority. It would also put massive pressure on his most moderate members in districts that voted for Biden if impeachment came to the floor. Like the two Trump impeachments, it is highly unlikely the Senate would vote to convict Biden given its Democratic majority and the supermajority vote needed to remove Biden. 

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However, at least one Republican in the Senate welcomed the prospect of more investigative tools for the House as it looks into the Biden family.

"I think we're gonna get to a point really soon, you may have to stand up a committee that then would open an impeachment inquiry just to do the investigation, particularly because the White House isn't cooperating at all," said Sen. Josh Hawley, R-Mo.

He added, "The American people deserve to know if the president's a crook."

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However, even Sen. Chuck Grassley, R-Iowa, who has been among the leading voices on investigating the president and Hunter Biden, said in a floor speech about evidence detailing a bribery allegation against the president that his focus is on federal law enforcement.

"I want to make clear what my oversight focus is and will be: holding the Justice Department and the FBI accountable to explain to the American people what they did to investigate and what they found," Grassley said.

Fox News' Brianna O'Neil and Ben Florance contributed to this report.