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Republicans have been slow-walking or outright blocking many of Mr. Obama’s nominees, but they signaled Thursday that they would be willing to relent and allow votes on most of them.

Mr. McConnell said the only nominees he was reluctant to approve speedily were two NLRB nominees and the consumer bureau nominee — each of whom Mr. Obama has installed to temporary appointments using his recess powers. A federal court ruled that the maneuver violated the Constitution.

It takes only a majority vote to approve a nominee on a final vote, but the minority is allowed to filibuster to prevent that final vote, and it takes 60 votes to end the filibuster.

The increasing use of filibusters — or even their threat — has shifted power from the president, whom the Constitution gives the right to nominate officers of the United States with the “advice and consent” of the Senate.

Mr. Obama tried to regain some of that power by using his recess appointment powers in an unprecedented way. He appointed members to the Consumer Financial Protection Bureau and the National Labor Relations Board even though the Senate didn’t consider itself in recess.

Two federal appeals courts have ruled that Mr. Obama’s NLRB appointments violated the Constitution. The Supreme Court said last month that it would hear one of the cases, and the ruling could dramatically circumscribe the president’s power.

Mr. Reid waited until after he cleared a major immigration bill through his chamber to raise the filibuster issue, knowing it would be so contentious that it could have ruined his chances to win the bipartisan vote on that bill.

Republicans contemplated using the nuclear option in 2005 to end Democratic filibusters of judicial nominees, but backed off after a bipartisan “Gang of 14” senators struck a deal that preserved the filibuster in limited circumstances.

That deal held for several years but has since dissipated.

Not all Democrats are onboard with Mr. Reid’s move.

“My position has been the same since the Republicans tried to jam judges in 2005 and we stood up against it and said, ‘You shouldn’t be changing the rules by breaking the rules and almost every Democrat spoke against it,’” said Sen. Carl Levin, a senior Democrat from Michigan.

But with 54 Democrats in the chamber, Mr. McConnell said, he expects Mr. Reid to be able to find the 51 votes he needs to force the rules change.