Mr. Issa, Republican of California, adamantly opposes letting the Postal Service touch the retirement fund surpluses. All of the other bills involve such a step, which Mr. Issa has described as a “multibillion-dollar bailout funded by the taxpayers.”

Representative Gerald E. Connolly, a Virginia Democrat who has proposed an overhaul bill, contested Mr. Issa’s characterization, calling the use of the word “bailout” politically charged and “disingenuous.”

“This is their money,” he said of the Postal Service.

The regulatory commission and the Office of Personnel Management, which manages the retirement funds, agree with the Postal Service’s position that it has been overpaying. Estimates of the surpluses range from $50 billion to more than $80 billion.

The popularity of Mr. Issa’s bill among House Republicans, many of whom campaigned against policies labeled as bailouts, may be the deciding factor in whether a major overhaul is approved this year. Representative Dennis Ross, the Florida Republican who heads the House oversight subcommittee that works on postal issues, is so far the only co-sponsor on Mr. Issa’s bill. He said he did not support withdrawals from the retirement funds but stopped short of calling the other proposals bailouts.

So far, 14 House Republicans and 161 Democrats are listed as co-sponsors of Mr. Lynch’s bill.

Despite cutting $12 billion in costs over the last four years and eliminating more than 200,000 career positions in the last decade, the Postal Service is expected to lose $8 billion for the second consecutive year as mail volume has dropped by about 20 percent from 2006 to 2010.

“There’s no disagreement with anyone that we will continue to reduce head count,” Mr. Donahoe, the postmaster general, said. “We will continue to do network consolidations and route consolidations.”

Last month, to keep the organization afloat, Mr. Donahoe stopped making required payments of $115 million every other week to one retirement fund. Without legislation, the Postal Service will again run out of money in the fall, and Mr. Donahoe said he would have to take more drastic steps.