D.C. City Council member Arrington Dixon, who is running for Council chairman, was endorsed yesterday by three of his council colleagues and Del. Walter E. Fauntroy, while a union representing some city policeman announced its support for the reelection of Mayor Walter E. Washington.

Council members David A. Clarke (D-Ward 1), Polly Shackleton (D-Ward 3) and Willie J. Hardy (D-Ward 7) joined Fauntroy in backing Dixon, saying that the Ward 4 Democrat would be the best person to represent the majority of the 13-member council and work effectively with other council members.

Dixon's major opponent in the Sept. 12 Democratic primary, maverick council member Douglas E. Moore, has frequently accused Dixon of being controlled by the city's business interests, especially landlords, realtors and developers.

But Clarke, who is also running for reelection and has often sided with Moore on the council, said the major concern in the campaign should not be how a given candidate might vote, but rather how effectively the candidate could run the council.

"We can say what we want in respect to individual issues," Clarke said at the press conference, held in the council chambers at the District Building. "My main forces is the orderly administration of the legislative processes of this city."

All three of the council members, as well as Fauntroy (D-D.C.) said they would work actively to get support for Dixon in their wards. In the past, Dixon has acknowledged having a limited city wide organization, a situation stemming from the fact that his previous two council campaigns have been for the seat representing Ward 4 in upper Northwest Washington.

Dixon said yesterday, however, that his campaign will not be concentrated in any ward of the city.

For Fauntroy, one of the most popular politicians in the city, the endorsement of Dixon completed what amounts to an undeclared slate for top city offices, headed by him and including, in addition to Dixon, Council Chairman Sterling Tucker, whom Fauntroy has endorsed for mayor.

Fauntroy said, however, that he considers the unified effort, a "partnership" rather than a slate. He said he does not plan to make any endorsements in ward or council at-large races.

Dixon has already endorsed Tucker for mayor, but had not received the endorsement of Dixon and said it represented "no public mandate."

"This is the same group that lined up against (former D.C. school superintendent) Barbara Sizemore (in the 1977 City Council election)," Moore said. "I would think they wouldn't think any differently about me because I, too, am about change."

The police group that endorsed Washington, the Policemen's Association of the District of Columbia, is not the official bargaining agent for city police. That group is the International Brotherhood of Police Officers, which announced in May that it is supporting council member Marion Barry for mayor. The two groups are now vying for representation as bargaining agent for the 4,100 police.

The group endorsing Washington represents some 3,500 members of the D.C. police, U.S. Park Police and the Uniformed Division of the Secret Service (Executive Protective Servi ce). About 750 of those members are D.C. policeman, and about 1,000 are actually city residents, according to Joseph S. Goldring, the organization's president.