WASHINGTON — A proposed compromise toward ending Saturday U.S. mail delivery is falling flat with unions while getting the support of a Senate committee chairman leading a push to advance stalled postal legislation.

Discussions have involved letting the U.S. Postal Service end Saturday mail delivery if mail volume or revenue drops below a specified level, Sen. Tom Carper, the Delaware Democrat who leads the Senate Homeland Security and Governmental Affairs Committee, said Thursday in an interview at Bloomberg’s Washington office.

“I’m attracted to the idea of a volume trigger that will incentivize postal employees to work harder, sell harder and incentivize mailers to mail more,” Carper said.

The idea is meeting resistance from postal unions, which want to keep Saturday delivery and the jobs that go with it. Legislation that would relieve the money-losing service of billions of dollars in annual health-care and pension obligations and allow other changes is stalled in both chambers of Congress.

Postmaster General Patrick Donahoe proposed ending Saturday delivery without congressional approval before his board overruled him in April, following an opinion by the Government Accountability Office that he lacked that authority. Congress first required mail deliveries six days a week in 1981.

The service is open to Carper’s idea, David Partenheimer, a spokesman, said in a phone interview.

“This is a very interesting and creative solution that Chairman Carper’s developed, and could be part of a comprehensive legislative package to resolve our financial situation,” he said.

A postal bill passed by the Senate last year didn’t allow an end to Saturday delivery, which has been part of proposed House legislation that hasn’t come to a vote in the full chamber in either of the past two years.

Jeanette Dwyer, president of the National Rural Letter Carriers’ Association, said her union opposes any idea that may end Saturday mail delivery.

“Bottom line is you ought to be talking about how to increase your business,” she said in a phone interview. “The Postal Service was never intended to be a business that made huge money. It was intended to be a service to the American public. That’s why it was created.”

The service narrowed its loss to $5 billion in its last fiscal year as it increased revenue for the first time since 2008. It’s testing same-day delivery for Amazon.com in two cities as part of efforts to boost revenue and volume.

Carper’s committee canceled a session scheduled for Wednesday to advance the Senate postal bill after deciding it needs more work.

The Senate needs to take the lead if any postal measure is to become law, Carper said.

“I’m not sure Darrell Issa can do a bill in the House,” because of opposition from rural Republican lawmakers who aren’t interested in cutting postal services, he said.

Issa, R-Calif., is chairman of the House committee that oversees the Postal Service. He sponsored a measure that passed his committee in July on a party-line vote.

The American Postal Workers Union, whose members work in mail processing plants, also opposes setting a threshold to end Saturday delivery, spokeswoman Sally Davidow said in an email. Unions are asking for relief from a congressional requirement that the service pay now for health-care costs of future retirees.

“The solution to the financial crisis is to eliminate the pre-funding requirement,” she said. “And there are many ways the USPS could increase revenue — licenses, basic banking for the unbanked, etc., without cutting service. Package volume is increasing and that trend is expected to continue.”