If the current scenario unfolds as expected, Sen. Patrick J. Leahy (D-Vt.) will walk into a joint Senate-House conference committee meeting today and deliver the fatal blow to the pet project of Washington's city officials and business community - the downtown convention center.

In doing so, the 37-year-old fresh man lawmaker will have emerged in eight months from the Ireative obscurrity of a back-row seat in the Senate to a prominent role in congressional manipulations of District of Columbia affairs.

It was only last February that Leahy was appointed to the Senate Appropriations Subcommittee on the District budget and promptly elected as its chairman, succeeding Sen. Lawton Chiles (D-Fal).

Although it is a post not cherished by ambitions senators seeking national influence, Leahy soon made it clear that he would take the job seriously.

Doing so led him into a bitter controversy over the convention center, which was approved last month by the House at the urging of its veteran District Appropriations Subcommittee chairman, Rep. William H. Natcher (D-Ky).

When the matter came before the Senate last Tuesday, Leahy took the opposite view, that the center should be rejected. After sharp debate, Leahy's view prevailed by a vote of 65 to 25. Leahy took most of the Senate's elders with him in the vote.

When debate ended, Senate Majority Leader Robert C. Byrd (D-W.Va.), himself a former chairman of the District Appropriations Subcommittee, took the floor to praise Leahy for "a masterful job" in managing his first bill through Senate passage.

For the city government, Leahy's views and the lopsided vote was a clear signal that the Senate no longer can be regarded - as it was by city officials for many years - as a sort of benign "court of appeals" willing to overturn adverse budgetary decisions recommended by Natcher and ratified by the House.

When the conference committee meets today to resolve differing House and Senate versions of the District budget for fiscal 1978 - and, in the process, the fate of the convention center - it is clear Leahy has the stronger hand to play. He is expected to control a majority of the votes to be cast by the committee's Senate menbers.

The convention center is not the only issue on which Leahy has jolted city officials. Since assuming the chairmanship he has:

Commissioned and distributed a Library of Congress study showing the District has the highest operating expense per resident of any American city of comparable size, even when the costs of state and county functions are figured in.

Questioned whether facilities for the new University of the District of Columbia are being overbuilt, and moved to prevent spending $57 million on construction until a detailed master plan can be adopted.

Recommended a 1978 federal payment to the city, in lieu of taxes on U.S. owned property, of $276 million - a full $20 million less than the House recommendation, and $24 million authorized by law.

Opposed Mayor Walter E. Washington's and the White House's recent proposal to authorize a federal payment of $317 million for the 1979 fiscal year, which will begin next Oct. 1.

Trimmed an earlier supplemental federal payment for 1977 by $2 million in order to pressure the city into more aggressive collection of parking tickets and bills due from patients for treatment at city-owned clinics and hospitals. This, Leahy explained at the time, reflects "my own brand of fiscal conservatism."

For such actions, Leahy has been sharply criticized by some city officials.

City Council Chairman Sterling Tucker found the cut in the 1977 supplement payment "an intrusion on our budget process, which I personally resent."

Council member Marion Barry Jr. (D-at large) was so outraged at the rejection of the convention center plan that he assailed Leahy as "this little rinky-dinky senator from a state like Vermont." After Leahy criticized Barry's comments at a Senate Appropriations Committee meeting, Barry sent a letter of apology.

A balding and once-overweight six-footer (he shed about 60 pounds this past year), Leahy does, in fact, represent a state that is small in population: 445,000 somewhat more than half of that District. It is one of but two states (the other is Wyoming) that containsno census-defined metropolitan areas.

Before being elected to the Senate by a razor-thin margin in 1974 to succeed the venerable George D. Aiken (R), who was retiring, Leahy was prosecuting attorney for the county that contains Burlington. Vermont's largest city (population 40,000)."

Leahy, in a recent interview, said he does not find representation of a small rural state an impediment in his senatorial work on the totally urbanized District. Among other things, he pointed to his own past residency in Washington.

A dozen years ago, he and his wife Marcelle lived in Washington's Glover Park neighborhood while he attended Georgetown University Law School and his wife worked as a nurse at the old Mount Alto Veterans Hospital. The oldest of their three children, Kevin, was born in Washington.

Their home here now is in McLean, in suburban Fairfax County - a location with dirt roads and many trees that was chosen, Leahy said, because it offers the chilldren surroundings not unlike those they were accustomed to in Vermont.

Leahy said he reads exhaustively about Washington - books, articles, documents, mail. When the District budget was being considered by his subcommittee, Leahy said he chose the names of some letter-writers at random and telephoned them to discuss their concerns. When the convention center was being actively considered, Leahy made two walking tours of the dilapidated three-block site, talking to merchants (who were divided in their attitudes toward it).

Ultimately, on the convention center, it became a matter of judgement, bound to raise questions not only about the project but also about the budgetary control Congress continues to hold over the District, even with the considerable home rule achieved by the city in 1975.

City officials contended the center would earn as much as $12 million a year in economic benefits for the city, chiefly in the form of new taxes, payrolls and commercial development around the center site. Leahy predicted the opposite: the project would lose $4 million to $9 million a year.

When Sen. Lowell P. Weicker Jr. (R-Conn.) asserted that Leahy was frustrating home rule, Leahy dissented.

"I would be perfectly willing to see total home rule given to the city," Leahy said, "but until that happens . . . the home rule charter requires us to enact a budget for the District of Columbia, and the law requires us to carry out that responsibility.

"The fact is that the center would be funded by a $110 million loan from the U.S. Treasury and paid back, in part, with federal funds that come to the city from federal payments."

President Carter has urged that the District be granted freedom from federal (including congressional) budgetary control by 1982, by which time both Natcher and Leahy may have moved on to other assignments, or left office.

Before full local budgetary control can be achieved, Leahy said, a new way must be found to set the level of the annual federal payment without requiring congressioal review of the city budget, item by item.