Frequent fliers on Eastern Airlines last week found an extraordinary letter in the mail -- an attempt by the company to "get our side of the story across" in a labor battle that will determine the fate of one of the nation's major airlines.

Although the letter from Eastern's president and chief executive, Phil Bakes, talked about Eastern's improved on-time performance and assured passengers of the airline's commitment to safety, the bulk of it was spent detailing the carrier's losses and "the need to fix our labor cost problem."

Eastern's long-festering feud with its labor unions comes to a head -- again -- at midnight this Thursday, when the airline's contract with the International Association of Machinists and Aerospace Workers expires.

The IAM, which represents 12,000 skilled mechanics, baggage handlers and other workers, is the prime target of Eastern's latest cost-cutting campaign, which began when the airline was acquired by the low-cost, largely nonunion Texas Air Corp. late in 1986. Within months Texas Air Chairman Frank Lorenzo called for a $490 million reduction in labor costs from all of Eastern's unions. Negotiations between Eastern and the IAM are stalled, and relationships are bitter, but under the terms of the Railway Labor Act -- which also governs bargaining in the airline industry -- no strike is imminent.

Both sides have asked for a federal mediator to help resolve their substantial differences, but no mediator has been named yet. If mediation fails, the National Mediation Board can propose arbitration. If the parties reject arbitration, the union is free to strike after a mandatory 30-day cooling-off period.

The letter from Bakes came during a period of almost daily reports that Eastern was considering steps as drastic as selling its profitable air shuttle operations to stem mounting losses. Eastern has declined to comment on these reports, which have been attributed to unidentified corporate sources.

But Bakes wasn't shy about discussing the carrier's financial problems in his letter to the frequent flyers. Noting that the airline's losses for the first three quarters of the year amounted to $67.4 million and had resulted in the layoff of 3,500 workers, Bakes noted that "losses are nothing new to Eastern."

The company has lost $500 million in the past 17 years, he said. "The lengthy saga of financial losses requires a permanent fix to the fundamental Eastern problem -- an inefficient labor cost structure."

If Eastern is unsuccessful in achieving that "permanent fix," the alternatives may include disposing of assets. Shortly afer Lorenzo acquired the airline, Eastern sold its computerized reservation system to Texas Air Corp. And earlier this year Eastern transferred six aircraft to Continental Airlines. Eastern has also been granted permission for Continental to fly some of Eastern's Latin American routes.

"I think selling assets or selling parts of the company is something they would consider doing if they had to," said Edmund Greenslet, an airline analyst with Merrill Lynch Research. "The critical time comes when the union is free to go on strike. Then we find out what's really going to happen." He said he doubts that the company "will do major surgery prior to that."

Charles E. Bryan, president of the IAM district that represents Eastern employes, said the threats to sell off pieces of the airline were part of a campaign by the company to put pressure on the unions. "It's intended to have the maximum psychological impact," Bryan said.

Eastern's unions said they believe they have a weapon more powerful than the strike, thanks to their dual role as both workers and shareholders in the airline. In negotiations with Eastern's previous management in 1984, the union agreed to $263 million in concessions and received 1.6 million shares of preferred stock in the company. The unions are convinced they can use these stock holdings to keep Eastern from selling assets, although Eastern management disagrees.

Bryan would not go as far to call the threat to sell assets a bluff by Lorenzo, however. "I would never suggest that anything he's doing is a bluff," he said.

Lorenzo has had repeated confrontations with airline unions beginning in 1983, when he cut short negotiations with Continental's unions by filing under Chapter 11 of the federal bankruptcy code, laying off workers and unilaterally reducing salaries. He also began New York Air, now part of Continental, as a nonunion carrier.

Nancy Currier, a vice president of the flight attendants union, also expressed the belief that the reports that Eastern might sell off some of its assets were part of a psychological game to force concessions. She said her union also was convinced that its stock holdings could prevent Eastern from selling any assets without union approval. She said that was why her union agreed to the concessions in its negotiations last year in exchange for the stock.

Although the flight attendants are not yet in negotiations, their relationship with management has been strained.

Eastern's relations with its pilots union have also been bitter. The company has experienced heavy turnover in its pilot force, and the pilots have launched a "Max Safety" campaign. The pilots said it was needed to focus attention on what they called a deterioration in maintenance standards, while Eastern management has denounced it as "a labor slowdown disguised falsely as a concern for safety." Eastern pilots are also trying to organize Continental's pilots into a union.

The two major fronts in Eastern's labor wars now are its negotiations with the IAM and the litigation front.

"I think the company is still very turnaroundable, but we have to make progress on our labor cost goals," said Bakes. "The most important element of that is the IAM. It's the biggest cost element by far."

The company proposes to reduce costs by what amounts to pitting the unskilled members of the IAM against the skilled mechanics in the union. Eastern has proposed large pay reductions for employes in the unskilled categories over the next three years. The pay cuts would be offset somewhat by an opportunity for those workers to be retrained as aircraft mechanics at Eastern's expense. The majority of the IAM members at Eastern are unskilled.

"We've offered to pay for retraining for those who want to be mechanics and also to pay their wage while going to school, which takes about two years," said Bakes. If the company gets the concessions it says it needs, then it can buy more airplanes and will need more mechanics, he said. "But we do need a labor structure that works before we press that rebuilding button," he said.

Bakes said the average compensation for baggage handlers at Eastern was $43,000 including benefits. Eastern officials said that American Airlines baggage handlers earn between $5.77 an hour and $14.78 an hour, compared with Eastern baggage handlers who earn from $9.32 to $15.60 an hour. Workers at Eastern reach top pay after seven years while those at American take 12 years. Some of American's baggage handling is being performed by a subsidiary, AMR Services, which pays beginning workers $4.25 an hour, Bakes said.

Eastern's fight with its unions is not confined to the negotiating table. Throughout much of the year, the two sides have been opposing each other in a series of court actions, many of them centering on the rights of employe shareholders.

Last week, a member of the the Air Line Pilots Association filed suit as a shareholder in Delaware state court challenging a "transfer scheme ... to move Eastern's valuable business and assets to Continental and other nonunion, low-cost subsidiaries of Texas Air, all to the detriment of both Eastern and its preferred stockholders."

Bakes called the latest suit a "ridiculous charge. That stuff is just part of the litigation-of-the-week strategy that the union has used." Bakes said that unions, acting as holders of preferred stock, cannot block the sale of assets as long as Eastern gets a fair price for them.

Earlier this year Lorenzo made a tender offer for the union-held stock. The unions have challenged the action and a federal court judge has temporarily blocked the tender offer.

Bakes said he would not predict the outcome of negotiations with the IAM. "There should be a very high probability of a negotiated settlement, but I'm not making a projection."

"If a man or woman landed here from Mars, they would say the chances of an agreement, of these two parties reaching agreement, was very, very high," said Bakes. "The measurable, objective situation is so very, very clear and very, very stark. We're not sitting here with an extremely profitable company in an extremely stable industry trying to get something more for the shareholders," he said, again invoking Eastern's losses.

But the process is complicated by personalities and the history of labor relations at the perennially troubled airline. Bryan once negotiated what was then hailed as a breakthrough labor agreement that traded concessions for stock and a broader role for labor, including a seat on the board for Bryan. The IAM leader still speaks wistfully about those days when Eastern was the model of "codetermination."

Now life is much different. Bryan is no longer on the board and has been rebuffed in attempts to deal directly with Lorenzo, a man he bitterly denounces. He accused Lorenzo of trying to "destroy the unions at Eastern and the quality of life for Eastern employes.

"Anything Frank Lorenzo can do to enhance his wealth, he's going to do," he said.