WASHINGTON — Senators today moved toward resolving their feud over Republican filibusters of White House appointees, hoping to avoid a Senate rules change by Democrats that would worsen the partisanship already troubling the chamber.

Officials said both parties were discussing a plan to 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.

The Senate voted 71-29 to clear the way for eventual confirmation of Cordray, who President Barack Obama installed when the Senate was in recess, angering Republicans.

Senate Majority Leader Harry Reid expressed confidence that a broader deal was within reach, even if it would leave open the possibility of future filibusters of Obama’s executive nominees. Reid had been threatening to change the rules, to bar such filibusters.

Senators are not questioning the ability to keep using filibusters — in which 41 of the 100 senators can block action — on legislation and judicial nominees, who seek lifetime appointments.

Under the proposal, 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.

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

If ratified, the deal would mark a retreat by Reid from his insistence on Monday that Republicans promise not to filibuster future executive nominees. Republican leader Mitch McConnell had privately offered to clear the way for the currently contested nominees — providing Block and Griffin were replaced — officials in both parties said. That’s largely the deal Democrats agreed to today.