The Air Line Pilots Association entered round-the-clock contract negotiations with Eastern Air Lines yesterday as a rival union claiming to represent strike-breaking pilots tried to stop those talks and force the pilots to choose which union they want to represent them.

Negotiators for ALPA and Eastern were prepared to work through the night in Washington under the auspices of bankruptcy court examiner David I. Shapiro in an effort to complete the first step toward restoring labor peace since the airline was wrested from the control of Texas Air Corp. Chairman Frank Lorenzo last month.

Martin Shugrue, the trustee appointed by a federal bankruptcy judge last month to run the airline, was said to be on his way from Miami to Washington yesterday to help work out a settlement that could put some striking ALPA pilots back to work.

In the meantime, the recently formed Eastern Pilots Association petitioned the National Mediation Board for an election to determine who should represent Eastern pilots. The new union claims to represent 98 percent of the 1,775 pilots working at Eastern -- all of them either ALPA members who crossed their own union's picket lines or pilots hired to replace striking workers.

Eastern was struck March 4, 1989, in a contract dispute with the International Association of Machinists, which represents the company's mechanics and ramp workers. The picket lines were honored by both the pilots and the flight attendants, forcing Eastern to seek bankruptcy protection within a week.

In petitioning for a new election, the new union called on Eastern to immediately suspend contract negotiations with ALPA until the mediation board acts. In a letter to Shugrue, union President Tom Cooper said his union would consider continued negotiations with ALPA an interference by the company in his union's election process. He also warned that federal labor law contains criminal penalties for such acts.

A spokesman for Eastern would not comment on the Eastern Pilots Association's action. He acknowledged that the airline was in negotiations with ALPA, but would not comment on the status of the talks.

Shugrue has insisted since his appointment that he plans to rebuild rather than sell the airline's assets piecemeal and that achieving labor peace is a major goal. However, Shugrue has not ruled out selling the entire airline to another party. Northwest Airlines has been rumored to be interested in buying Eastern.

The major issue in negotiations with ALPA are the terms under which the pilots would return to work. Eastern has said that it is interested in immediately hiring back 89 pilots of the approximately 1,400 that are still out of work.

Any election to determine union representation would include the 1,400 ALPA members who remain out of work and the 1,775 working pilots largely claimed by the rival union. ALPA is claiming in a federal court case in Florida that approximately 400 pilots currently on Eastern's payroll are not entitled to vote. Those votes could tip the balance in any election.

There were indications yesterday that the mediation board might postpone a decision on the union's election petition until it has decided a two-year-old proceeding in which Eastern's unions claim that Eastern and Continental Airlines, another subsidiary of Texas Air Corp., are legally one company. If the mediation board were to decide in favor of the unions, any elections would have to include Continental employees. A decision in that case could come within the next few weeks.

Negotiations between the IAM and Eastern are on hold pending the outcome of the ALPA talks.