Mayor Marion Barry, who has said he would like to enhance his reputation nationally, has directed D.C. officials to launch a long-shot bid to persuade the Democratic Party to hold its 1984 national convention at the new $100-million Convention Center.

The city already has spent $25,000 to retain a prominent Washington law firm and a public relations specialist to help make its presentation to the 27-member Democratic National Committee's site selection panel next month.

City agencies have been mobilized to assemble data to make a case that this is the best site for the convention, which could pump from $15 million to $20 million into the local economy.

"We're going to give it a good shot," Michael C. Rogers, the Convention Center's deputy manager, said yesterday. "I think we have a chance because the Convention Center building and the community can accommodate the convention."

"No other city can make the claim that every major hotel is on a public transportation route," Rogers added.

But Washington's chances of capturing the convention on its first try appear slim, according to some Democratic Party officials, partly because of mounting pressure within the party to hold the next convention on the West Coast or in the Sun Belt.

The last two Democratic conventions were held in New York.

Party Chairman Charles T. Manatt, a California businessman, privately is pushing to have the convention near San Francisco, while Robert S. Strauss, a former party boss and an influential Texas lawyer, seeks to have the convention in Houston.

Television network executives also may lobby to have the convention in Houston to minimize the cost of shipping their equipment between the Democratic convention and the Republican one, which is scheduled to be held that summer in Dallas.

In looking over the field of contenders, including New York, Chicago and Detroit, convention organizers also may be interested in choosing a site that adds a little spice and variety to the lives of the 20,000 delegates, alternates, party officials and newsmen who will attend.

"A lot of delegates are regulars to Washington and may be looking for a little variety," said one member of the Democratic National Committee's staff.

From a symbolic standpoint, however, Washington might be the perfect place for the Democrats formally to launch their 1984 assault on the Reagan administration, according to one high-ranking party official.

"There's a group, not limited to the District government, that feels that by coming to Washington we're making a statement about Reagonomics and the cuts in programs," this official said. "There's a tremendous story to be told about what Reagan has done to the federal government, and to the country as a result."

For the present, party officials will say publicly only that none of the contenders, including Washington, is being ruled out.

"They Washington are making a serious effort and, at this stage, everyone is in the running," said William Sweeney, deputy chairman of the Democratic National Committee.

The city has retained the law firm of Stroock, Stroock & Lavan and the public relations firm of Ernest Wittenberg Associates to assist in preparing its bid application, which is due Jan. 28, and in making a formal presentation to the site selection committee in mid-February. The committee will make its decision on April 8, after visiting each of the sites.

Barry and Theodis R. Gay, chairman of the D.C. Democratic State Committee and a member of the site selection committee, will host a reception at the Convention Center on Feb. 3 for members of the Democratic National Committee to lobby for support. Barry recently lunched with Manatt, also in an effort to drum up support.

Washington's Convention Center, which opened for business this month, has four exhibition halls, 40 meeting rooms and more than 800,000 square feet of public space. Convention Center officials say that about 37,000 hotel rooms are available for conventioneers in the greater metropolitan area, including 18,000 in the city itself.

"The city has an excellent chance; it's so logical it escapes most people's thinking," said Bruce Kirschenbaum, a member of the Stroock law firm who represented New York City when it successfully bid on the 1976 Democratic national convention.

Kirschenbaum contends that regional party rivalries count for less today than in the past in deciding where to stage a national convention.