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(CNN) —  

Senate Republicans on Monday both praised and expressed skepticism about the decision by House Democratic leaders to vote on an impeachment resolution, saying they hope it will open up the process and provide fairness to President Donald Trump.

At the same time, some GOP senators strongly doubted that the House Democrats are prepared to do anything short of impeaching him.

“It’s about time,” declared Sen. David Perdue, a Republican from Georgia and close ally of Trump’s who said he wants to study the resolution to see if “it’s really going to provide due process” to the inquiry that’s been underway for several weeks.

Republicans also warned that so much damaging information about Trump has “selectively leaked” out of the closed-door depositions in the House impeachment probe that a “vote now is a bit like un-ringing a bell,” according to Sen. Lindsey Graham, a Republican from South Carolina.

GOP senators, who are would-be jurors in an impeachment trial, took credit for forcing Speaker Nancy Pelosi to vote on the resolution after Senate Majority Leader Mitch McConnell and Graham, who chairs the Judiciary Committee, unveiled a resolution last week criticizing how House Democrats have handled the inquiry, pressing them to hold a floor vote to formally initiate it and to allow Trump to call witnesses on his behalf and Republicans to issue subpoenas.

“Today, they were forced to change course,” Graham said.

While some House Republicans – and the White House – complained that the Pelosi resolution was too little, too late, the second-ranking Senate Republican had a different take, saying it “would be an entirely different conversation” and “it would be great” if the House “would open this process up.”

“I suspect that it would create an entirely different environment,” said Sen. John Thune of South Dakota, “if the House was playing ball in terms of trying to make this a fair process where everybody has due process under the law.”

Republican Sen. James Lankford of Oklahoma said, “This process wasn’t right. If they can clean the process up, that sure does help a lot just for us to be able to see things.”

“It’s long overdue,” added Sen. Susan Collins, a Republican from Maine who is up for reelection next year in a purple state. She is one of just three Senate Republicans who did not co-sponsor the McConnell-Graham resolution.

Sen. Mitt Romney, a Republican from Utah and frequent critic of Trump’s explained why he also didn’t sign on to the Senate impeachment resolution.

“I’ve been reluctant from the beginning trying to advise the House on how they should proceed,” Romney said. “Now that the speaker has scheduled a vote on the impeachment inquiry, I think the issue has been over taken by events.”

Sen. John Cornyn, a Republican from Texas, said the McConnell-Graham measure, which was not yet scheduled for a vote, has “done some of what we intended: It forced an open process.”

Cornyn, however, expressed frustration that House Democrats have only one thing in mind: impeachment.

“We know where this is headed; we might as well get it over with,” Cornyn said. “They’ve already made up their minds, so let’s get it on.”

McConnell declined comment to CNN about the developments.

Republican Sen. John Kennedy of Louisiana argued it may be too late to change the rules of the inquiry and that the Democrats have decided already Trump is guilty.

“Any modifications that they’re making right now is just like adding paint on rotten wood,” Kennedy said. “I think they’ve made up their mind. If they’ve made up their mind, they need to go ahead and send the articles (of impeachment) to us.”