Anyone who wants to run for mayor of the District of Columbia has to talk to people like Flaxie Pinkett.

Pinkett is a realtor with important ties to the business community. She was recently named "Man of the Years" by the Greater Washington Board of Trade, and she has considerable influence among an important segment of established, middle-class black voters. The last time around, she backed former mayor Walter E. Washington, considered out of the running in next year's primary. So all the politicians who would like to be elected mayor in 1982 have been trooping over to see Flaxie.

"Seven or eight of them have talked to me," Pinkett said recently. "I told the last one that if another one came, I'd run myself."

Pinkett added, "I just told them that it's too early for me to be on anybody's team."

It is about time for the jockeying and the deal-cutting to begin in anticipation of the 1982 mayor's race. But the three most frequently mentioned potential contenders -- incumbent Marion Barry, 45, former City Council chairman Sterling Tucker, 57, and at-large council member John Ray, 38 -- are finding few of the city's politicians and business leaders willing to commit themselves.

Each of the three, who are the most visible in an extensive cast of Democratic hopefuls, has liabilities. These minuses, plus the possibility of a host of other potential candidates, have left allegiances and intentions far less obvious than at the same point before the 1978 election.

By this time in 1977, the two main challengers to Washington, Barry and Tucker, had emerged and were jockeying for the support of D.C. Del. Walter E. Fauntroy, whose political organization was considered the best in town.

Barry was busy on the cocktail party circuit trying to establish himself as a credible candidate, while Tucker was trying to become viewed as heir apparent to the throne.

Part of the difference between May 1977, and May 1981 might be that this time there are so many potential candidates.

Among those at least considering a run for mayor are Council Chairman Arrington Dixon, 38, council members Betty Ann Kane (D-At Large), 39, John A. Wilson (D-Ward 2), 37, David A. Clarke (D-Ward 1), 37, and Charlene Drew Jarvis (D-Ward 4), 39, as well as former University of the District of Columbia board chairman Ron Brown, now counsel to the Democratic National Committee, and former Secretary of the Army Clifford L. Alexander, who lost to Washington in 1974 and now is on the lecture circuit.

To varying degrees, all have made the rounds of the businessmen and politicians who can provide the necessary money and campaign help to make a credible run in 1982 for an office that could take $500,000 to win. Generally, they have been told their efforts are premature.

Barry was a slim winner in the 1978 Democratic primary. But as the incumbent, he has assumed a clear and early front-runner status. He has presided over the city government 2 1/2 turbulent years, in which budgets have been curtailed and demand for city services has risen.

A succession of budget-related crises has cost Barry political support. In December, a poll taken by The Washington Post found that only three of every five persons interviewed who said they voted for Barry in 1978 said they would vote for him again.

There is also uncertainty about the effectiveness of Barry's efforts to build support among the older, more moderate middle-class blacks who rejected him in 1978. He has also lost some support among the younger and more affluent whites who strongly backed him three years ago.

Tucker, who after the 1978 race landed as an assistant secretary of Housing and Urban Development in the Carter administration and now heads his own consulting firm, has been out of the local spotlight for nearly three years.

Dixon and Jarvis each now lay claim to Tucker's old base in the well-off neighborhoods of Ward 4 just east of Rock Creek Park, and Ray has designs on Tucker's supporters to the east in Ward 5.

Moreover, Tucker's defeat was generally credited to lack of organization, and there are few indications that he has held together what organization he had.

Some of Tucker's former supporters say they are unwilling to join him for another attempt at gaining the mayor's office.They believe Tucker has had his chance.

"I would say that my present inclination is that I will probably support the mayor," said businessman and Democratic National Committeeman John Hechinger, a staunch Tucker backer in the last election.

Democratic State Committee Chairman Robert B. Washington Jr. has also told friends that he does not plan to support Tucker again.

Tucker's campaign ended bitterly, with recriminations over campaign workers' paychecks and blame-laying for the defeat. This baggage, say former Tucker supporters, has persuaded many of them to stay clear of him in 1982.

Ray, a relative newcomer who burst onto the scene when he ran for mayor in 1978, has faced only token opposition in citywide contests. "What I hear people saying is 'John Who?'" said one active Democrat, a member of a rival camp, who asked not to be named.

And Ray's aggressiveness on the council -- including blasts at Barry, his former political patron who was instrumental in helping Ray win his council seat in 1978 -- has given ammunition to critics who charge that he is overly ambitious, even presumptuous, in seeking the mayor's office at this stage of his political career.

At stake is the highest elective office in the city at a time that could be crucial to the District, as sluggish revenues and Reagan administration budget cuts are expected to force the city to trim back services even further.

City employees, some of whose support was important to Barry in 1978, will be seeking to regain the parity in salaries with federal workers that they lost last year when Barry gave them a 5 percent pay increase instead of the 9.1 percent that went to the federal work force.

A still unanswered question is where Washington's former supporters will go. Tucker and Ray have both been courting the Ward 5 and Ward 7 blacks who made up Washington's base of support, spending their Sundays in the churches of Northeast Washington. But so far, few commitments.

In addition to the reluctance of business leaders to commit themselves yet to a candidate, the interest by so many council members in the race has kept alliances among politicians from solidifying.

In 1978, Fauntroy was another focus of political power in the city, and his endorsement was eagerly courted early in the campaign.

But Fauntroy's candidate for mayor in 1978 (Tucker) lost, as had his candidate for mayor in 1974 (Alexander), and his candidate in last year's Ward 7 City Council race (Johnny Barnes). As a result, the still-popular Fauntroy has developed a reputation for having no coattails, and there has been no intense scrambling for his endorsement.

Several Barry allies said the mayor has told them he definitely plans to run again. But they say he has not yet begun an intensive drive to use his status as an incumbent to get major campaign contributors from the business community to commit themselves. This will eventually be a key to the mayor's strategy, supporters say.

"If you're talking about the average person in Washington who is active politically and is an active giver, to start approaching people with the election so long away isn't going to work," said a lawyer who raised money for Barry in 1978.

One factor, the lawyer said, is uncertainty over whether Ray, Tucker or any of the other hopefuls will actually run. "They're having lunches with people," the source said, "but I don't think anybody's putting money on anybody running."

Tucker has approached people like Pinkett and lawyer R. Robert Linowes, a former Board of Trade president, in search of support. But those he has approached say they have told him it is too early to commit themselves, and some of his stalwarts from 1978 say they plan to sit out a 1982 Tucker campaign, if it develops.

Ray has approached businessmen like Luther H. Hodges, who heads the National Bank of Washington, in search of support. Hodges said he talked with Ray and came away "impressed," but made no commitments.

Like Barry and Tucker, Ray has made no public announcement that he intends to run for mayor.

At a birthday party in his District Building suite last week, Ray's friends and supporters chanted "Eighty-two! Eighty-two!" and gave him a birthday card that said: "To the Leader of the Pack."

The leader depicted on the card was an elephant, prompting Ray to say, "I may be slow, but I'm out there."