WASHINGTON — The Senate stepped away from the brink of a meltdown on Tuesday, clearing the way for confirmation of one of President Barack Obama’s nominees long blocked by Republicans, nearing a deal to fill several other vacancies and finessing a Democratic threat to overturn historic rules that protect minority-party rights.

“Nobody wants to come to Armageddon here,” said Sen. Chuck Schumer, the New York Democrat whose talks with Arizona Republican John McCain were critical in avoiding a collision that had threatened to plunge the Senate even deeper into partisan gridlock.

McCain, a veteran of uncounted legislation struggles, told reporters that forging the deal was “probably the hardest thing I’ve been involved in.”

There was no immediate response from the White House, although Democratic senators said the terms of the compromise were acceptable to the administration.

Under the agreement, which both sides were reviewing at midday, several of seven stalled nominees would win confirmation quickly, including Labor Secretary-designate Tom Perez; Gina McCarthy, named to lead the Environmental Protection Agency, and Fred Hochberg to head of the Export-Import Bank.

Even before the agreement was ratified by the rank and file, Richard Cordray’s long-stalled nomination to head the Consumer Finance Protection Bureau advanced toward approval on a test vote of 71-29, far more than the 60 required.

Two nominees to the National Labor Relations Board, Richard Griffin and Sharon Clark, were to be replaced by new selections, expected to be submitted by President Barack Obama later Tuesday and steered toward speedy consideration by Senate Republicans. Obama installed Griffin and Clark in their posts by recess appointments in 2011, bypassing the Senate but triggering a legal challenge. An appeals court recently said the two appointments were invalid, and the Supreme Court has agreed to review the case.

The seventh nomination at issue, Mark Pearce’s selection to a new term as NLRB chairman, was relatively uncontroversial, and is likely to be approved along with the replacements for Griffin and Clark.

“I think we get what we want, they get what they want. Not a bad deal,” said Senate Majority Leader Harry Reid, D-Nev.

There was more to it than that.

Scarcely 24 hours earlier, Reid had insisted that if Republicans didn’t stop blocking confirmation of all seven, he would trigger a change in the Senate’s procedures to strip them of their ability to delay. At the core of the dispute is the minority party’s power to stall or block a yes-or-no vote on nearly anything, from legislation to judicial appointments to relatively routine nominations for administration positions.

While a simple majority vote is required to confirm presidential appointees, it takes 60 votes to end delaying tactics and proceed to a yes-or-no vote. Reid’s threat to remove that right as it applied to nominations to administration positions was invariably described as the “nuclear option” for its likely impact on an institution with minority rights woven into its fabric.

The same term was used when Republicans made a similar threat on judicial nominations in 2005 — an earlier showdown that McCain helped defuse when it was his own party threatening to change the rules unilaterally.

As part of the deal over Obama’s nominees, Republicans agreed to step aside and permit confirmation of several, some of whom they had long stalled. Cordray was first appointed in July 2011, but a vote was held up by GOP lawmakers who sought to use his confirmation as leverage to make changes in the legislation that created his agency.

McCarthy was named to her post in March, and Republicans dragged their feet, demanding she answer hundreds of questions about the EPA. At one point, they boycotted a committee meeting called to approve her appointment.

Perez, also nominated in March, is a senior Justice Department official, and was accused by Republicans of making decisions guided by left-wing ideology rather than the pursuit of justice.

As described by officials, the deal is strikingly similar to a proposal that Senate Republican Leader Mitch McConnell floated in remarks on the Senate floor last week during an unusually personal exchange with Reid. At the time, the Kentucky Republican also said he had told Obama last January to drop his hopes of confirmation for Griffin and Clark and instead name two replacements. He relayed the same message again last month to Vice President Joe Biden, a former senator with whom he has a long relationship.

Tuesday’s developments unfolded the morning after a closed-door meeting of nearly all 100 senators, many of them eager to avoid a rules change that could poison relations between the two parties at a time the Senate is struggling in an era of chronic gridlock. About three dozen lawmakers spoke in the course of a session that lasted more than three hours, and while few details have emerged, several participants said later it had been a productive meeting.

Sen. Susan Collins, R-Maine, said she had urged others to “look ahead and think about the time when we would have a Republican president with Republican Senate and there could be someone appointed who was completely unacceptable to my Democratic colleagues and was nominated to run their favorite program” She said she asked if they “really want to give away their right to filibuster that individual.”

Sen. Richard Durbin, D-Ill., said the sense of history hung over the meeting, which was held in the Old Senate Chamber, where lawmakers had debated slavery and other great national issues for much of the 19th century. “Senator McCain talked about Webster, Jefferson and Madison. We knew that we were on sacred political ground,” he said.

McCain told reporters that with McConnell’s knowledge, he had been involved in talks for several days in search of a compromise, speaking with Biden, White House Chief of Staff Denis McDonough and numerous senators.

“At least 10 times it came together, and then fell apart because there’s always some new wrinkle,” he said.

Associated Press writers Alan Fram, Charles Babington, Donna Cassata, Josh Lederman and Sam Hananel contributed to this story.