The House was expected to vote as early as Thursday on a proposal by Rep. Patrick Murphy, a Pennsylvania Democrat who served in the Iraq war, that would repeal the 1993 law known as "don't ask, don't tell."
The legislation - a compromise struck with the White House and agreed to by the Defense Department - would give the military as much time as it wants before lifting the ban.
Under the bill, the president, defense secretary and chairman of the Joint Chiefs of Staff must first certify that the new policy won't hurt the military's ability to fight.
"We need to get this done, and we need to get it done now," said Murphy.
Also as early as Thursday, the Senate Armed Services Committee was expected to take up an identical measure, proposed by Sens. Carl Levin, D-Mich., and Joe Lieberman, I-Conn.
As in the House, the Senate provision would be tucked into a broader bill, authorizing hundreds of billions of dollars for the troops, that is expected to win broad support.
Supporters said this week the Senate panel had enough votes to pass the bill after key holdouts, including Sen. Ben Nelson, a Nebraska Democrat, announced they would swing behind it.
"In a military which values honesty and integrity, this policy encourages deceit," Nelson said.
Nelson said a provision in the bill giving the military the power to decide on the details of implementing the policy was key to his support because it "removes politics from the process" and ensures repeal is "consistent with military readiness and effectiveness."
Advocates hoped the momentum in the Senate would carry over to the House, where several conservative Democrats - including Rep. Gene Taylor of Mississippi - threatened to oppose the massive defense spending bill if it included the repeal provision.
Defense Secretary Robert Gates has said he supports repeal but would prefer that Congress wait to vote until he can talk to the troops and chart a path forward. A study he ordered is due Dec. 1.
"With Congress having indicated that is not possible, the secretary can accept the language in the proposed amendment," said Pentagon press secretary Geoff Morrell.
The service chiefs this week urged the Senate panel not to vote until the Pentagon could complete its survey of military personnel.
"The value of surveying the thoughts of Marines and their families is that it signals to my Marines that their opinions matter," Marine Commandant James Conway wrote in a letter to Sen. John McCain of Arizona, the panel's top Republican.
Adm. Mike Mullen, the nation's top uniformed officer and chairman of the Joint Chiefs of Staff, told graduating Air Force Academy cadets on Wednesday that they need to support a changing military.
Mullen didn't speak directly about the "don't ask, don't tell" policy. But the chairman, who has said that the policy unfairly forces troops to lie, said service members should question convention.
"Few things are more important to an organization than people who have the moral courage to question the direction in which the organization is headed and then the strength of character to support whatever final decisions are made," Mullen said.