D.C. City Council Chairman Sterling Tucker is narrowly leading his two chief rivals, Mayor Walter E. Washington and City Council member Marion Barry, among registered voters in the Sept. 12 Democratic city mayoral primary, according to a citywide Washington Post poll.

Tucker had 24 percent to 20 for Washington and 18 for Barry in the poll. But the survey also shows that with three months of campaigning to go, the outcome of the race is much in doubt. The largest group of voters - 35 percent - said they are undecided.

The results of the poll, in which 1,020 people were interviewed by telephone or in person between June 1 and 5, suggest a change in voter sentiment since November, when a Post survey showed Tucker holding a wide lead among voters polled after casting their ballots in a school board election.

The new poll shows significant disenchantment among a large segment of voters with the performance of Mayor Washington, the first person elected as the city's chief executive under the limited home rule granted by Congress in 1973. About 30 percent of those interviewed said they would not vote for Washington.

However, the poll also shows the mayor holding a hard core of support, with his backers the least likely to switch to another candidate. He is favored by voters with family incomes of less than $18,000, older voters and voters whose education did not continue past high school.

Barry far outpolls his rivals among white voters and among voters 18 to 45 years old. He trails badly among black voters.

Tucker has the broadest base of support, according to the poll, and is regarded by voters somewhat more warmly than Barry or Washington. He places ahead among voters who went to college and those with an annual family income of $18,000 or higher.

Tucker said he was pleased that the poll showed him still ahead and that he was not concerned that the second Post poll seems to indicate his support is eroding.

But he noted that his declining percentage coincided with an increase in the number of undecided voters.

"They didn't go to Marion or the mayor," Tucker said. "They went to undecided. If they'd gone to the other candidates, it would bother me. "I've got to go after hte soft Tucker vote.

"They're waiting for me to come after them," he said. "It's up to Setrling Tucker to sell himself to the public."

Barry said the poll results did not surprise him. "When I announced my candidacy I said I was the underdog," he said. "I've always said I was four, five points behind Sterling Tucker.

"When people see the other side of Sterling Tucker and his lack of leadership, they'll turn to me," Barry said. "Ultimately, the race is going to be between Walter Washington and me."

He said his substantial support among white voters is "primarily because I'm working in areas (on the City Council) where white people and middle class blacks would pay more attention - taxes and money, efficiency in government, not the delivery of social services." Barry is chairman of the council's finance and revenue committee.

"The only poll that counts is the poll on election day," the mayor said through his campaign manager, Lacy Streeter. Reached later, the mayor declined further comment "until I have a chance to analyze the figures," but he said he did not know when that would be.

Disenchantment with the mayor was reflected by several measurements in the poll. For one thing, only 6 percent said they would not vote for Tucker and 13 percent said they definitely would not vote for Barry, compared to the 30 percent who said they would not vote for Washington.

In addition, the poll showed that only one-third of those who said they voted for the mayor in his victorious 1974 race against Clifford ,. Alexander plan to do so again, at least as of now. One-third of the former Washington supporters have switched to the two council members running for mayor, with Tucker winning twice as many of them as Barry. The other one-third of the mayor's former supporters say they are undecided.

Many of those polled by Post reporters said the mayor had not lived up to their expectations when they voted for him in 1974, while people who voted for Alexander four years ago overwhelmingly said they found not reason to switch to the mayor this time.

"He's had his chance to perform." said a 47-year-old federal government worker who lives in the Brightwood community of Northwest Washington and voted for Alexander in 1974. "It's time for new blood."

Father Russell L. Dillard of St. Anthony's Catholic Church, who voted for Washington in 1974, said simply: "Though he's done an adequate job, it may be time for him to step aside." The priest said his vote this year will go to Tucker.

One of the poll's chief findings was the importance voters assign to the position of mayor. Those interviewed were asked whether they felt it made much difference who the mayor was - or whether each in a series of 13 city problem areas would remain the same regardless of who was mayor.

Wide majorities said a mayor could make a difference on almost every issue mentioned, and only on one - the problem of keeping down the cost of living in the city - did a majority say a mayor could not make much difference.

Those polled said Tucker would be the best at handling 10 of the 13 problems, including such things as improving public schools, curbing crime and "providing efficient responsive city services." Barry was viewed as best able to fing jobs for young people and "keeping poorpeople from being displaced from their neighborhoods by real estate speculation." The mayor tied with Tucker in one category, as being best able to maintain "good relations between blacks and whites."

Some of those polled did not always vote for the candidate they thought would do the best job at handling various issues. One such person was William L. Torrence III, an environmental health officer for the city's Department of Human Resources.

Torrence said Tucker would do better at a variety os issues, but he nonetheless is going to vote for Barry. "I think if they give him a chance," said Torrence, "they'll find he's a good powerful politician, very capable."

Samella T. Anderson, a technician with the National Archives, said she will again vote for Washington, despite the fact that she thinks the other candidates can do better at solving various city problems.

The mayor, she said, is "the best balanced overall."

The mayor also ranked third behind Tucker and Barry on a so-called "feeling thermometer." Those polled were asked to state their warmth or coolness toward the three candidates and other public figures in the District, by assigning ratings from zero to 100.

By this gauge, Tucker had an average of 65, Barry 57 and Washington 54.

In The Post's November survey, Tucker led with 39 percent to 23 for Barry and 20 for Washington. The two polls are not exactly comparable, however, because the earlier on was conducted among what might be called "hard core" voters, people whose views were tapped as they left polling places after voting in a school board election.

Those interviewed in the new Post poll were selected from among registered Democrats who had a record of voting at least once in the past 10 elections - making them closer to the electorate likely to vote in the Sept. 12 primary.

Voters in the recent survey who said they did not know who they planned to vote for were asked toward whom they are leaning. When these voters were added to those who said they had made up their minds, Tucker got 28 percent and both Washington and Barry 22 percent.

With a three-way race, some of the voters have partially made up their minds. One such voter is Fred Bradley Jr., a 37-year-old construction worker who voted for Alexander in 1974.

He said he knows that he will not vote for Washington, but is undecided btween Tucker and Barry. At least one of the determining points, he said, is which candidate will do the most for inner city minorities in the construction industry.

Tucker, Washington and Barry are only three of the eight announced candidates in the Democratic primary, but The Post's poll shows that at least so far the five other candidates have had a negligible effect on the race.

The names of Dorothy Maultsby and John L. Ray were included in the survey, but they go only 1.4 and .3 percent of the vote, respectively. Voters were given an opportunity to name other people as their choice for mayor, but only 1.1 percent did so. The other announced Democratic candidates are Charles S. (Trummie) Cain, Richard A. Jackson and James Clark.