WASHINGTON — Republicans and Democrats reached for compromise Tuesday over stalled Obama administration nominees, and Senate Majority Leader Harry Reid expressed confidence a deal was in sight.

Officials said the two sides were discussing a plan that would permit prompt confirmation for most of the contested nominees, including Tom Perez to head the Labor Department. Gina McCarthy to run the Environmental Protection Agency and Richard Cordray to lead the Consumer Financial Protection Bureau.

At the same time, President Barack Obama would drop efforts to win confirmation for two members of the National Labor Relations Board and name two replacements who would receive speedy consideration. Richard Griffin and Sharon Block were originally named to their posts as recess appointees, meaning they bypassed confirmation. An appeals court has since ruled their appointments were invalid, and the Supreme Court has agreed to review the case.

The agreement would not resolve deep partisan divisions over the future use of filibusters to block a president’s executive nominees, said officials on both sides. The officials who described the talks did so only on condition of anonymity, saying they were not authorized to disclose details.

“It is a compromise, and I think we get what we want, they get what they want. Not a bad deal,” Reid said on the Senate floor.

The developments unfolded on the morning after a closed-door meeting of nearly all 100 senators, eager to avoid a showdown over the rules for confirmation that could poison relations between the two parties.

If ratified, the deal would mark a retreat by Reid from his insistence on Monday that all seven of the pending nominees be confirmed.

Republican Leader Mitch McConnell privately offered to clear the way for several of the nominees, officials in both parties said.

The Senate was scheduled to hold several test votes on the matter around midday. Top Democrats had been negotiating with a rump group of Republicans, including Sen. John McCain of Arizona.

If a compromise breaks down, there could be big ramifications for politics and policymaking for years to come.

Standing alone, the rules change that Reid proposes is limited. It would end the ability of 41 senators, in the 100-person chamber, to block action on White House nominations other than judges.

The out-of-power party still could use filibuster threats to block legislation and judicial nominees, who seek lifetime appointments.

But critics say Reid’s plan would be likely to prompt Republicans to retaliate by doing even more to reduce the minority party’s rights if the GOP regains control of the Senate. That could happen as early 18 months from now, after the 2014 elections.

“It’s a decision that, if they actually go through with it, they will live to regret,” McConnell has said.

Unlike the 435-member House, the Senate has a long and bumpy tradition of granting rights to minority-party members. The most powerful tool is the filibuster, which can kill a measure by using endless debate to prevent a yes-or-no vote.

The mere promise of a filibuster can block Senate action on almost anything unless 60 of the 100 senators vote to overcome it. Filibuster-proof majorities are rare, and Republicans now hold 46 Senate seats.

Both parties have accelerated their use of the filibuster threat in recent times. Since Obama took office in January 2009, Republicans have threatened filibusters repeatedly, infuriating Democrats.

Reid said Lyndon B. Johnson faced one filibuster during his six years as Senate majority leader. In the same length of time as majority leader, Reid said he has faced 413 threatened filibusters. The tactic, he said, blocks action on routine matters that Congress once handled fairly easily.

Democrats acknowledged that Republicans will turn any such rules change to their advantage if they regain the Senate majority, which the two parties have often swapped in recent decades.

White House spokesman Jay Carney told reporters the Senate “needs to confirm this president’s nominees in a timely and efficient manner.” That will be true, he said, “for the next president, and the next president after that. This has become ridiculous.”

Senate Republicans particularly objected to two union-backed members of the National Labor Relations Board, Griffin and Block.

Republicans also had opposed Cordray, Obama’s pick to lead the Consumer Financial Protection Bureau, created in a Wall Street oversight revision that Republicans opposed.

Asked Monday if Obama worries that a filibuster rule change would make the Senate even more dysfunctional, Carney said, “Well, it boggles the mind how they would achieve that.”

This notion that things can’t get much worse in the often stalemated Senate seems to have convinced numerous senators and interest groups in recent months that there is little risk in changing traditions to end at least some of the logjams.