Senate Majority Leader Mitch McConnell (R-Ky.) is signaling to colleagues that he will loosen his grip on floor debate to quell mounting frustration within the Senate GOP over the paltry number of votes he has allowed on amendments.

McConnell, who last week became the longest-serving GOP Senate leader in history, is seeking to avoid another blowup over the issue given the tensions within his ranks.

Sen. Bob Corker (R-Tenn.) accused McConnell and Senate Republican Whip John Cornyn (Texas) in a private meeting Wednesday of blocking his amendment to limit President Trump's ability to impose tariffs, according to sources familiar with the meeting.

His office also sent out a press release Tuesday that accused "Senate leadership" of blocking an up-or-down vote on his tariff amendment.

Corker is far from the only Republican senator to complain about the issue.

In recent days there has been growing pushback from rank-and-file Republican senators over what they see as an effort by leaders to stymie debate.

"I just believe put a bill on the floor and let people start amending it. And if somebody wants to object to an amendment, they need to object and record their objection and stand up and be counted. If somebody wants to offer an amendment, they should be able to offer an amendment," said Sen. John Kennedy (R-La.).

He said floor debate and consideration of amendments have been too constrained this year.

"I've made that position known to my colleagues," Kennedy said.

Asked if there should be more votes on amendments going forward, Sen. Susan Collins (R-Maine), a prominent moderate, said "yes."

And Sen. Rand Paul (R-Ky.) accused his leadership of being "complicit" in preventing debate on amendments.

"If they wanted to allow amendments they can at any time," he said. "Yeah, they're complicit."

McConnell promised in 2014, when Democrats still controlled the Senate and then-Majority Leader Harry Reid (D-Nev.) kept a tight rein on the debate, that Republicans would open the floor to more amendments if they were in charge.

"Voting on amendments is good for the Senate, and it's good for the country. Our constituents should have greater voice in the process," he said in a January 2014 speech about how Republicans would run the Senate differently if in the majority.

He argued that constituents should "know where we stand on the issues of the day, regardless of whether the majority party thinks those issues are worth debating or voting on."

GOP leaders pushed back at Corker this week.

Cornyn in the conference meeting said that he was not blocking the Tennessee senator's amendment on tariffs, even though he told reporters publicly that he didn't think it wise to pick a fight with Trump over the issue.

But McConnell also has signaled that he wants the debate on upcoming appropriations bills to be more open so various GOP colleagues can get votes on their top-priority amendments, according to GOP lawmakers.

"I think Mitch has crossed over. He's a survivor," said a Republican senator, who predicted that McConnell will work to ensure more amendments get votes in the future.

"He understands there's a lot of frustration," the lawmaker added.

This lawmaker also said Corker was misdirecting his blame for the matter, claiming that it was Sen. James Inhofe (R-Okla.), the acting chairman of the Senate Armed Services Committee, and not McConnell, who blocked Corker's amendment.

McConnell and Cornyn have argued that they don't control what amendments get voted on because it requires unanimous consent from all 100 members of the chamber to schedule a vote on a particular amendment.

Cornyn told reporters earlier in the week that leaders would tell colleagues that Republican senators have to stop blocking each other's amendments.

A GOP leadership aide said that McConnell doesn't control the floor debate, but instead only "manages" it because he needs consent to set up votes on amendments.

McConnell has carefully managed debate on the Senate floor to avoid exposing rifts among his Republican colleagues or picking fights with President Trump.

He didn't want to risk Corker's amendment finding it's way onto the defense bill because it could have drawn a veto threat from the president.

Trump argued to GOP senators at a White House meeting earlier this month that he needs to retain his power to impose tariffs so he can use it as leverage in negotiations with Canada and Mexico over the North America Free Trade Agreement.

Another Republican senator, however, said McConnell and Cornyn prefer having few amendments debated and voted on because it gives them greater control over legislation.

"As long as we don't have amendments, they're in control and they love being in control," said the lawmaker.