A group of Senate Republicans helped pass legislation to ban workplace discrimination against gay employees on Tuesday, piling pressure on leaders in the House of Representatives to rethink their opposition.

The Employment Non-Discrimination Act, known as Enda, is aimed at extending national workplace rights for gay, lesbian, bisexual and transgender employees that are similar to those in place to prevent racial or sexual discrimination.

It received bipartisan support in the Senate on Tuesday afternoon, passing 64 votes to 32 after an amendment to exempt religious groups secured the backing of an increased number of moderate Republicans.

But the legislation is expected to join the a growing list of Senate bills languishing in the House of Representatives after speaker John Boehner said he was against allowing a vote, arguing the legislation was unnecessary and could prove too expensive and litigious for businesses.

"The speaker believes this legislation will increase frivolous litigation and cost American jobs, especially small business jobs," said a spokesman for Boehner on Monday, when it first become clear the bill was likely to receive sufficient support to proceed in the Senate.

The White House welcomed the final Senate vote on Thursday which it said was an important step in the two-century-old "story of more citizens realising the rights and freedoms that are our birthright as Americans” but joined the angry calls for a vote in the House.

“One party in one house of Congress should not stand in the way of millions of Americans who want to go to work each day and simply be judged by the job they do,” said President Obama.

Jim Inhofe of Oklahoma, one of the Senate Republicans to vote against Enda, claimed the measure was "pandering to the special groups that I think should not have to be singled out by themselves. I think they're normal citizens like everybody else."

But 10 Senate Republicans, led by John McCain, chose to vote for the legislation after negotiating to expand the Enda exemption in ways that would make sure employers with religious affiliations were not penalised by the government in any way.

The amendment was written by Republican senator Rob Portman and co-sponsored by Kelly Ayotte, Dean Heller, Orrin Hatch, and McCain. They were joined by Pat Toomey and Susan Collins, who said it was “long passed time to fill the gap” in protections for the LGBT community.

Speaking ahead of the Senate vote, Democrat Dick Durbin called on Boehner to remember the civil rights tradition of the Republican party, which was founded by anti-slavery activists in 1854.

“Our history books are littered with those public figures who say we can't end discrimination based on race, discrimination based on disability, discrimination based on gender,” he said. “I call on John Boehner to make sure he doesn't get on the wrong side.”

A number of large national employers and states already have anti-discrimination policies in place, but an estimated 29 states have no protections for gay and lesbian employees and 31 offer nothing to protect transgender staff.

New York Democrat Charles Schumer railed against what he claimed was short-sighted obstructionism by House Republicans, who have also rejected a bipartisan immigration bill passed by an even bigger majority in the Senate.

“The problem is that House of Representatives has become a legislative blackhole, a broken cog in our legislative wheel,” said Schumer. “Every time we pass a bill, it's like we're banishing it to a faraway jail.”