The Defense Department has asked President Reagan to issue an executive order that could, in effect, make the Pentagon the final arbiter in nearly all disputes between the military and the labor unions that represent U.S. civilian employes at overseas military bases.

The order could prevent the unions from appealing certain grievances to the Federal Labor Relations Authority--a move that the unions claim would effectively destroy their ability to represent 10,000 union members employed overseas by DOD.

DOD, while acknowledging yesterday that it sent a proposed executive order to the Office of Management and Budget in February, refused to discuss it. A spokesman for OMB said the proposal is being circulated among federal agencies for comment.

A copy of DOD's proposed order was given to The Washington Post by the National Federation of Federal Employees. While that union is not the only one that represents DOD employes overseas, its demands in the Republic of Korea apparently prompted DOD to appeal to Reagan.

Relations between the federation's Local 1363, which represents about 300 of the 1,100 DOD civilian employes in Korea, and Gen. John A. Wickham Jr., commander of U.S. forces in Korea, became strained more than two years ago when the union demanded that Wickham begin negotiating which products are rationed at military post-exchanges (PXs) and the number of vehicles that employes can bring duty-free into Korea.

According to a DOD letter to Budget Director David A. Stockman, Wickham refused to bargain about those two restrictions because they are part of Status of Forces Agreements (SOFA) between Korea and the United States. The restrictions are necessary, DOD said, to limit black-market trading in Korea. Allowing unions to negotiate rules covered by SOFA agreements would be the same as allowing a union to negotiate a diplomatic agreement with Korea, DOD reasoned.

When Wickham refused to bargain, the union appealed his decision to the FLRA, which ruled in favor of the union in 1980. The arbitrators said PX goods and duty-free vehicles were "conditions of employment"--job benefits offered by the military to attract employes and, therefore, were negotiable under the Federal Service Labor-Management Relations statutes. DOD immediately appealed that decision to a U.S. appeals court, which has not yet rendered a decision.

At the same time, Wickham asked DOD to draft the executive order for Reagan. In the request, DOD Deputy Secretary Frank C. Carlucci says the union problem in Korea could "exacerbate relationships" between the two countries and, therefore, poses a national security threat. Carlucci recommended that Reagan cite national security reasons and suspend federal statutes that require DOD to negotiate with the union.

"It is hard to believe that the union's right to bargain over mayonnaise, shampoo, pepper, crock pots, hair dryers, tennis rackets and so on threatens the national security," says Don C. Terrill, president of the NFFE unit in Korea.