Sen. Minority Leader Mitch McConnell, R-Ky., said on Tuesday he supports a full impeachment trial for Department of Homeland Security Secretary Alejandro Mayorkas over his handling of the border crisis.

"I think that would be the best way to go forward," McConnell told reporters after the weekly Senate GOP leadership press conference.

Top GOP leader behind McConnell, Republican Whip Sen. John Thune, also called for a full impeachment trial, while more GOP lawmakers argue Democrats are attempting to short circuit a trial. 

"The House of Representatives has determined that Secretary Mayorkas has committed impeachable offenses. That issue will come before the United States Senate. I believe the Senate needs to hold a trial," Thune announced. 

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Sens. Chuck Schumer and Mitch McConnell

US Senate Majority Leader Chuck Schumer (L), Democrat of New York; and Senate Minority Leader Mitch McConnell, Republican of Kentucky. (Saul Loeb)

"I’m going to make the argument — and I’m sure the Democrats will try to dismiss it — that we ought to be having a trial, conducting a trial of the United States Senate to determine whether these are impeachable offenses," he went on.

Senate conservatives have been putting pressure on Republican leadership to push forward with a trial as it's unclear whether the Democrat-controlled Senate will move forward with the House's articles. 

Sen. Rick Scott, R-Fla., is one of the lawmakers pushing for a full trial. He told Fox News Digital Tuesday that "we have got to make sure we have a real trial and get to real facts about what's going on here."

"So if he thinks he's done such a great job come and explain it. But what they want to do is they want to short circuit this," he said. 

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Sen. Rick Scott speaks during a news conference

 At center, Sen. Rick Scott (R-FL) speaks during a news conference with members of the House Freedom Caucus at the U.S. Capitol November 29, 2023 in Washington, DC.  (Drew Angerer/Getty Images)

Scott, alongside nearly a dozen Senate Republicans, signed a letter led by Sen. Mike Lee, R-Utah, last week urging McConnell "ensure that the Senate conducts a proper trial, and that every senator, Republican and Democratic, adjudicates this matter when the Senate returns."

"According to multiple briefings by your staff, Majority Leader Schumer and Senate Democrats intend to dispense with the articles of impeachment by simply tabling both individually," Lee wrote. "This is an action rarely contemplated and never taken by the U.S. Senate in the history of our Republic."

Meanwhile, Sen. Majority Leader Chuck Schumer, D-N.Y., didn't give clarity on Tuesday about whether a trial would be scheduled. After the House voted to impeach Mayorkas earlier this month, Schumer assured a trial would proceed this week. That now appears to be unlikely as a partial government shutdown looms. 

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Mayorkas

Homeland Security Secretary Alejandro Mayorkas is expected to face a House impeachment vote. (Getty Images)

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"Look, as I’ve said, the impeachment of Mayorkas is absurd. There’s not one drop of evidence that leads to any kind of charge of impeachment, and we’re going to handle it in the best way possible," Schumer told reporters on Tuesday.

There is also growing frustration among the 11 House impeachment managers about when they will present their articles to the Senate in the case of Homeland Security Secretary Alejandro Mayorkas. 

House managers serve as "prosecutors" and argue the impeachment case before the Senate

One impeachment manager told Fox they had been given "no clear guidance" about the roles they might play or when the House may even transmit the articles to the Senate. 

Fox News' Chad Pergram contributed to this report.