When the Department of Defense announced this year that it planned to establish a lucrative software research center at one of the nation's universities, officials from the University of Maryland quickly enlisted the 10 members of the state's congressional delegation to help put together a proposal to land the multimillion-dollar project near College Park.

At about the same time, Carnegie-Mellon University in Pittsburgh hired a former congressional liaison to President Reagan as well as lobbyists from eight Pittsburgh corporations to persuade the Pentagon to locate the institute at its campus in Pennsylvania.

Also in the bidding was Texas A & M University, which has the support of Republican Sen. John Tower, the powerful chairman of the Armed Services Committee, who wrote to the Pentagon to make the case for his state.

This week the Department of Defense is expected to announce where the $6 million center will be located, ending months of intensive lobbying and competition among a dozen universities and congressional delegations that see the potential bounty in having the Pentagon's chief software institute on their turf. Capitol Hill insiders say the list of universities that originally applied has been narrowed to Maryland and Carnegie-Mellon, which has resulted in new levels of rhetoric on both sides.

"Maryland is lobbying everybody, pulling all of their politicians in and calling up all these people," complained an aide to Sen. John Heinz, a Pennsylvania Republican who has visited White House deputy chief of staff Richard Darman and two top defense officials to push for Carnegie-Mellon. "We wouldn't dream of that."

The Pennsylvanians argue that they need the institute to help diversify a fast-eroding industrial economy and combat double-digit inflation.

To that, a Maryland congressional staff member said, "You can't take an unemployed steelworker and make them a software engineer."

Just last week, Maryland Reps. Steny H. Hoyer, a Prince George's County Democrat, and Marjorie S. Holt, an Anne Arundel County Republican, phoned Undersecretary of Defense Richard D. DeLauer to remind him of the "technical merits" of Maryland's proposal, namely the university's computer sciences department and the region's numerous high-technology firms.

These phone calls followed others made by the delegation and a letter sent by the 10 members last spring to Defense Secretary Caspar Weinberger.

Virginia, loath at times to support Maryland projects, in this case has responded to its neighbor's request for aide and brought out its Republican heavy hitters. Virginia's 10 House members and two senators signed a letter that Republican Sen. John W. Warner, a ranking member of the Armed Services Committee and former Navy secretary, delivered by hand to the Pentagon.

Although Maryland officials play down their efforts, they readily admit they are eager to secure the software center.

"It's a plum for sure, and we think we made a strong case for this being a key area for attracting the best in the world," Robert Smith, a University of Maryland vice president said, last week. "I am absolutely confident it was the best proposal."

So fierce is the competition that even some corporations have been drawn into the fray and are divided in their allegiances. One example is the Westinghouse Corp., which has a branch in Anne Arundel County supporting the Maryland proposal while a branch in Pittsburgh is pushing Carnegie-Mellon.

Unlike the computer hardware industry, which is located primarily in California's Silicon Valley, there is no single facility or region that serves as a headquarters for the newer field of software technology.

The Pentagon conceived of the Software Engineering Institute as a cost-saving measure that would bring together 250 engineers to explore software applications in defense weaponry, particularly in ships, planes and satellites. After a $6 million expenditure during the first year on the institute, which this year won congressional approval, the Defense Department would invest another $103 million during the next five years.

Maryland officials and politicians estimate that the institute would create about 25,000 jobs in the region. It would be the centerpiece of the Maryland Science and Technology Center in Bowie, scheduled to open in 1987 at the intersection of routes 50 and 301.

Maryland officials say their proposal should be at the top of the Pentagon's list because of the unversity's proximity to Washington, high-tech and defense firms, major transportation systems, and an area that offers a high quality of life.

The presence of the nation's first major software institute would give Prince George's County cachet it now lacks among its more elite neighbors, particularly Montgomery and Fairfax counties, leaders in recruiting high-tech firms.

"I don't think there is any doubt that, from Prince George's standpoint, it will be a major coup to have a facility of this size and magnitude here," Hoyer said last week. "It will be great for the university, great for Prince George's, and great for Maryland."

In the meantime, Maryland and Pennsylvania officials say they hope the final decision will not be based on political considerations.

"I know for a fact it will be made strictly on merit," the aide to Pennsylvania's Heinz said recently. "We really think Carnegie-Mellon has the best offer and the best bid."

"Hopefully it won't be made on a political judgment," Hoyer said. "On the technical merits, we know we're number one."