D.C. City Council member Marion Barry is under increasing pressure not to challenge Council Chairman Sterling Tucker for mayor next year. The pressure is coming from Tucker supporters and other city Democrats who fear a destructive split in the party's ranks and an undesirable precedent in a Tucker-Barry faceoff.

Del. Walter E. Fauntroy, a strong Tucker supporter, and other leading city Democrats have been urging Barry (D-at large) for the past several months to run as Tucker's heir-apparent in a unified Fauntroy-Tucker-Barry slate.

Several well-placed Democratic sources said last week that it is only a matter of time before Fauntroy, who is believed to have the largest political following in the city and is beginning to be viewed as the major king-maker in the city's Democratic Party, publicly endorses Tucker for mayor. In addition to his long-time friendship with Tucker, the sources said, Fauntroy's choice is aimed at establishing a precedent for political tradition in the still unsettled milieu of local Democratic politics.

"Tucker's older, he's more experienced and he's served his time as chairman. It's not Fauntroy that's against Marion. It's logic. It's common organizational sense," one source close to Fauntroy said. "We're getting to the point where we can no longer afford a popularity contest in every election. We want to build a system by which you get the best experienced people for the job."

Mayor Walter E. Washington continues to keep his future plans closely guarded, even from some of his most loyal supporters who, according to one source, want very badly to know soon what Washington intends to do.

For most of the past year the mayor has virtually been written off by leading city Democrats, influential businessmen and even some civic groups who contend that Washington's 10 years in office have been marred by bureaucratic scandals and punctuated with inefficiency and inaction.

In the past several weeks, however, some businessmen appear to be re-examining Washington's political work due in large part to continuing doubts about Barry and Tucker as genuine alternatives. And the mayor's supporters, when questioned by a reporter last week, he said, "Don't count the mayor out yet."

Many key city Democrats are afraid that if Barry, 41, does not run for Council chairman, Council member Douglas E. Moore, the maverick Democrat who has already announced his candidacy for Council chairman, will slip in as Tucker's successor because of his strong campaigning ability.

One Democrat close to Fauntroy said, Moore is "a nut, but he strikes the right chords. He's against gambling. He's for guns. He's against homosexuals and against the convention center."

In the face of the increasing pressure Barry asserted last week that he has ruled out any candidacy next year for any office except mapor. "I've rejected running for chairman of the Council in 1978," Barry said in an interview. "I'm trying now to qualify as a candidate for mayor."

Barry is continuing a hectic effort to line up possible contributors and campaign workers with a view toward persuading others that he can run a viable mayor's campaign with or without Fauntroy's support.

The tensions among the city's leading Democrats have become so strong, one source said, that Barry agreed to attend a recent meeting of most of the Democrats on the City Council only if there was minimal discussion of whom the majority of the city's Democrats would support for mayor and Council chairman in 1978. The meeting was convened by Fauntroy in an effort to begin forging a unified Democratic Party ticket for next year's elections.

While Tucker, 53, the former Washington Urban League director, is still considered the front-running Democratic mayoral hopeful, several sources said last week that his political prospects have been clouded by continuing allegations that he may have violated city restrictions on outside employment by the Council chairman.

Barry, of whom the business community - the key source for bankrolling campaigns - has always been somewhat leery, rekindled old fears recently when he called the chairman of the Senate District Appropirations Subcommittee "a rinky-dink senator" for rejecting plans for a $110 million civic center in downtown Washington.

The former street corner activist later apologized to Sen. Patrick J. Leahy (D-Vt.) for the remark. But some businessmen have not forgotten it that easily "That 'rinky-dink senator' statement really got the Hill riled up," one leading businessmen said. "People are wondering if Marion's calm enough to be in that position (mayor) without shooting from the hip."

"Before, you'd forget about the mayor. But it isn't that way any more," said a ranking member of the Metropolitan Washington Board of Trade. "But (now it seems) you've got defects on the other side (the Council as well).

"At least he (the mayor) was there, as compared to someone who has problems or who we have some questions about. Maybe we should take another look."

Not all businessmen share that view of the mayor, however. One banker, noting that the Washington had received business support in 1974 on the grounds of being a known quantity, said he thought such a rationale would not work again. "That's what the businessmen said last time," the banker said. "Now the business community ain't too swift, I'll agree. But they ain't that dumb, either."

There are frequent one-on-one meetings, a smattering of audiences with special interest groups and strategy sessions with key supporters, but the mayoral campaign maneuvering now adays is mostly a subterranean effort due to several factors.

In Tucker's case, the problem is a lawsuit filed in June in D.C. Superior Court by Corporation Counsel John R. Risher Jr. The suit contends that Tucker's working as a lecturer at Howard University for two years violated restrictions on the home-rule charter that limit the outside employment by the Council chairman. In addition, Risher maintains that the Howard post, which Tucker has since left, was used by Tucker to, in effect, get additional salary for his regular Council job, which Risher says is a violation of the city's conflict-of-interest code.

Risher has asked the court to remove Tucker from office for allegedly violating the charter provision, but no decision has yet been reached.

Sources close to Tucker are divided on what effect the outcome of the law-suit could have on his political future. One confidant of the Council chairman suggested last week that even if Tucker is ousted from office by the court, he could gain politically.

"The word on the street is that Sterling hasn't done anything really wrong, it's kind of like being guilty of spitting on the street," the source said. "So even if he loses, it could come out a plus, because here you'd have Sterling Tucker, the establishment candidate, becoming a victim of the establishment."

Others, primarily businessmen, believe that Tucker could be irreparably harmed by court-ordered removal from office. Initially, among businessmen, Tucker drew sympathy because it appeared that Risher, a close confidant of the mayor's, was carrying on a political crusade, one source said. But continuing allegations - even though not proved - are beginning to taint Tucker's image in the eyes of some.

Another factor that is keeping the politicking at a low-key level is a nearly year-old investigation by the U.S. Attorney's Office here of the financial relationships between millionaire developer Dominic F. Antonelli Jr. and Joseph P. Yeldell, a close confidant and top aide to Mayor Washington.

The federal investigators are trying to determine whether Yeldell, while serving as director of the city's Department of Human Resources, made several decisions that benefited Antonelli financially in return for $54,500 in personal loans. The mayor himself has appeared before the grand jury.

Until the grand jury closes its investigation, the mayor's political future is considered uncertain and, some of his advisers say, he is reluctant to make his political intentions known.

In the case of Barry, the caution against announcing his plans is the result of a strange quirk in the city's election law - plus, one source said, the obvious wisdom of being the only one of the three most frequently mentioned candidates not involved in a highly publicized legal controversy.

Free of legal battles and the burdensome chore of trying to chair the City Council, Barry has been the most active campaigner of the three. However, city law prevents him from declaring his candidacy for mayor without giving up his seat on the Council (his term does not end until 1980). But Barry has done everything legally possible to stay within those limits and still test his candidacy.

Some persons close to Barry say his desire to run for mayor is matched by his drive not to run for Council chairman. Privately, Barry has told acquaintances that he thinks the job of trying to make order out of an often chaotic 13 member Council is not worth the effort and basically a powerless job.

In addition, one leading Democrat noted, the present strength of the Council chairmanship is based in large part on the perceived weakness of the incumbent mayor.

"Sterling as mayor would be very different than Walter Washington as mayor. Sterling could very well go behind him (Barry as chairman) and start developing his own majority on the Council," the source said.

While Tucker, Barry and Washington have occupied the political spotlight in the mayor's race maneuvering, other names are constantly tossed around in political circles for both mayor and Council chairman. But none of them has announced firm intentions.

The Rev. David Eaton, pastor of All Souls Unitarian Church, has had a draft committee meeting for months, but is not expected by some to run for mayor or City Council chairman. H.R. Crawford, a Southeast Washington real estate manager who served as assistant secretary for Housing and Urban Development in the Nixon administration, is still mentioned as a possible candidate for either post, more likely for Council chairman.

The newest name suggested as a possible candidate for mayor, supported by some former backers of now Secretary of the Army Clifford Alexander, who unsuccessfully challenged Washington in 1974, is Ronald Brown, chairman of the board of the University of the District of Columbia.

Among those mentioned for Council Chairman have been School Board president Therman Evans, Councilmen Arrington Dixon and John WIlson, and the Rev. A. Knighton Stanley. Some expect that former school supt. Barbara Sizemore may seek a Council seat, possibly ward five in Northeast.