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Minority Leader Mitch McConnell, who took to the Senate floor for a second day to debate Mr. Reid personally, later told reporters that Mr. Reid’s move was akin to throwing “a bomb into the Senate, have it blow up and have everybody mad as heck.” He said the problem isn’t the rules but rather Mr. Reid’s inability to control the chamber.

“What we need is a majority leader with a different view about the Senate, consistent with its norms and traditions,” the Kentucky Republican said.

Two years ago, Mr. Reid resisted these kinds of rules changes, arguing that minority-party rights were too important to sacrifice for expediency. Instead, he reached a gentleman’s agreement with Mr. McConnell: He would allow more amendments to be debated if Mr. McConnell agrees to stop some of the Republican filibusters.

But that agreement crumbled, with each side blaming the other, and Mr. Reid said he probably made a mistake in not pushing the changes back then.

This year’s debate also marks a reversal from 2005, when Republicans held the Senate and Mr. McConnell supported the majority-vote method, nicknamed the “nuclear option,” to change filibuster rules for judicial nominees. Mr. Reid opposed it.

Republicans ultimately didn’t go through with that rules change after a bipartisan group of 14 senators struck a deal to curtail filibusters and approve more judges.

Today, the roles are reversed.

“We ought to be negotiating. Rules changes ought to be proposed by the majority leader and the minority leader,” Mr. McConnell said.

Mr. Reid said he is open to a counterproposal from Mr. McConnell if he has one.

“I’d be happy to talk to him about that,” Mr. Reid said.

Other senators also are calling for compromise.

In an op-ed column this year in The Washington Post, Sens. Carl Levin, Michigan Democrat, and Lamar Alexander, Tennessee Republican, broached the idea of allowing a bill to come to the floor for consideration without a 60-vote majority in exchange for restricting amendments to those relevant to the bill.

In a September floor speech, Mr. Levin repeatedly called for “self-restraint” and said his support to limit filibusters on the motion to proceed with a bill would continue regardless of which party controlled the chamber after the election.

He also said, however, that using the majority-vote method would be a disastrous change for the Senate.

“My frustration with the recent abuses of the rules does not overwhelm my duty to defend the uniqueness and integrity of this great institution,” he said.