Inflation and bickering among local governments are threatening to delay completion of the 101-mile Metro system well into the 1990s and drive the final price of the project to $10 billion, $2.5 billion more that is presently available.

Metro and local government officials agree that conclusion is inescapable after a study of Metro's latest construction cost estimates.

The problems come just five months after President Carter signed legislation that was supposed to guarantee the federal money to finish the subway at a total cost of $7.5 billion. That dollar figure already has been exceeded: Metro's newest inflation fed estimate is $8.2 billion with a 1990 completion date.

Further, if Metro is forced by federal and local budget pressures to continue building at a relatively slow pace of $325 million per year, and if inflation continues at only 10 percent, completion of the system could be delayed well into the 1990s. In that case, the $10 billion figure could be conservative, Metro officials said.

"It's ridiculous for us to estimate at that level because they'd run us out of here," one Metro expert said.

Metro, when laid out in 1968, was projected to be completed by now for a cost of $2.5 billion.

The other problem -- bickering -- has been with Metro to one degree or another since the original subway route structure was nailed to the wall in a fractious public meeting in 1968.

Metro's fragile coalition of two states and seven local governments plus the District of Columbia has held together remarkably well over the years, particularly when confronting a common enemy such as a federal demand to restudy whether to build the uncompleted segments of the system. When the members of that coalition have to deal with each other, however, it is a different matter.

Right now, these local governments are trying to agree on how to spend a limited amount of money for the next two years -- fiscal 1980 and fiscal 1981, and they've been unable to reach agreement for months.

There are several issues, but one is overriding: Prince George's County is afraid that Virginia will get all of its Metro lines first, then refuse to continue contributing to construction north of the Potomac River.

Francis B. Francois, a Prince George's County Councilman first and a Metro board member second, has been known to say in private sessions with other board members that "We've got to stop Virginia." In an interview on Thursday, Francois said, "It's not a matter of mistrust; it's just a general concern."

The construction schedule calls for the last Virginia construction money to be obligated in 1984 and the last Virginia section of tract to open for riders in 1986. That still leaves at least four more years before the last Maryland line is opened -- in Prince George's County.

Under Virginia law, local and county governments cannot obligate themselves to pay for something beyond their terms in office unless the voters approve that obligation in a referendum. No referendum has been scheduled.

Another possibility would be for the state of Virginia to guarantee that it will pay or back up local Metro construction costs, just as Maryland has done. That would require another major effort by Northern Virginians in Richmond and could take a few years.

"I just do not believe there is any way ultimately that Virginia will not play," said Irving G. McNayr, the retiring executive director of the Northern Virginia Transportation Commission. "Virginia has to. There is going to be a lot of soul-searching and fussing, but when the chips are down, they will have to play."

There is another string in Maryland's bow, however. It is generally agreed that some day, somehow, Virginia must add a Tysons Corner-Reston-Dulles Airport spur to the planned Orange Line extension to Vienna. If Maryland is to participate or agree to that spur -- as it must under the complex law that founded Metro -- Virginia will have to seek Maryland's help.

Richard S. Page, Metro's general manager, said last week that he was "dismayed that we have not made progress in 13 months" in determining -- one that has something for everybody -- Metro cannot apply for the federal money that has been budgeted. Time is not yet critical, but it is getting short.

Some progress was reported yesterday in resolving the problem after a fractious meeting Wednesday night and some preliminary attempts to make up on Thursday. Like all Metro crises, this one probably will be worked out.

It is at this point that the two problems -- inflation and bickering -- merge. If inflation slows a little bit, and if Metro can persuade Congress and the U.S. Department of Transportation to expand the amount of money they will permit each year for construction, the final price tag could be brought back to that $7.5 billion. If that happens, regional consensus will be easier.

Those are big ifs. Every year the program continues, Metro has a big bill for construction management and other non-construction items that just eat up money that could be buying concrete.

Another major factor is that Metro must divert a big chunk of its construction money -- about $158 million -- in 1981 and 1982 to purchase subway cars to run on new subway track.

The construction schedule for the first 65 miles is holding relatively firm. Thirty-three miles now are in operation. The 65-mile total will be reached late in 1983 with the openings (in sequence) of the Blue Line to Addison Road in Prince George's County (late this year); the Red Line extension from Dupont Circle to Van Ness Cener (late 1981); the Blue Line from National Airport through Alexandria to Huntington (early to mid-1982); the 14th Street Bridge crossing between the Pentagon and L'Enfant Plaza (late 1982), and the rest of the Red Line from Van Ness Center all the way out to Shady Grove in Montgomery County (late in 1983).

What happens after that is under debate.

"What troubles me," Page said "is that I don't see any evidence that any of the local and state governments have failed to make their payments in Metro history. There's no federal threat now. The feds are saying, 'Get your act together and put up your money.'"