Former D.C. City Council chairman Sterling Tucker, who just two weeks ago entered the race for his old job, is narrowly leading incumbent Arrington Dixon in the race for the Democratic nomination, according to a new poll by The Washington Post. Council member David A. Clarke is running a distant third.

The Post poll of 1,251 D.C. Democrats who said they were registered to vote in the Sept. 14 primary found 31 percent saying they supported or were leaning to Tucker, 28 percent for Dixon and 16 percent for Clarke. A large number, 25 percent, said they were undecided.

All three candidates ran almost equally among white voters, while Tucker was a slight favorite over Dixon among blacks, with Clarke well behind. The poll indicated no significant differences in voter preference between men and women.

Tucker, who narrowly lost a 1978 bid for the party's nomination for mayor and abandoned a second try for that office earlier this year, led in several other categories, however, including voters aged 45 and over, those with household incomes below $15,000 and those who worked for the D.C. government.

Of the city's eight wards, Tucker led in four--Ward 2 and Ward 6, in the inner city and Capitol Hill areas, Ward 5 in upper Northeast Washington and Ward 8 in far Southeast Washington. Clarke was the favorite in Ward 1, which includes the Adams-Morgan and Mount Pleasant neighborhoods. He has represented that ward on the Council since 1975.

Dixon led in Ward 4 in upper Northwest Washington, which he represented for four years. Tucker and Dixon ran about evenly in Ward 3, the area west of Rock Creek Park, with Clarke again running third. Thirty-five percent of those interviewed in Ward 3 said they were undecided.

In an effort to measure the feelings of voters toward the candidates, The Post asked respondents whether their opinion toward each candidate was favorable or unfavorable.

Tucker was regarded favorably by 46 percent of the respondents and unfavorably by 24 percent, while 30 percent said they had no opinion. Dixon's favorable-unfavorable rating was 42 percent to 26 percent with 32 percent having no opinion.

Twenty-seven percent had a favorable opinion of Clarke, while 17 percent were unfavorable. An unusually large number--56 percent--said they did not know Clarke well enough to have an opinion, suggesting that Clarke is not well known by a large segment of the electorate.

Tucker, who has been out of elected politics for nearly four years, said yesterday that the poll showed "the people are ready for me to return. For those who say it is too late, it clearly is not . . . . "There is a vote out there waiting for someone to come and get it."

Dixon, who was elected in 1978 when Tucker ran for mayor, said, "We expected that the poll results for Tucker would be unnaturally high. People have fond memories of a public official who has not been in the public eye for a while." Dixon said his own polls also showed a high undecided vote. "We expect to get it," he said.

Clarke questioned the poll results, noting that the survey, taken between June 17 and June 22, was done the same week Tucker announced his candidacy--an announcement widely reported in the news media.

"Where I am known, I got a good rating," Clarke said. "If it becomes a name-recognition campaign, I've got a lot to overcome. If it is an issue campaign, I can win."

At this early point in the campaign, the poll indicates that Tucker is initially getting some of the grass-roots support sought by Clarke, who has said much of his campaign is directed at poor and moderate-income residents.

In Ward 8 in far Southeast, which includes some of the city's poorest neighborhoods, Tucker was the choice of 41 percent of those interviewed, compared with 27 percent for Dixon, 7 percent for Clarke and 25 percent undecided.

In Ward 7, the far eastern tip of the city that includes several other low- and middle-income areas, Dixon was favored by 33 percent, Tucker by 31 percent and Clarke by 10 percent with 29 percent undecided.

Citywide, among voters in households with annual incomes of $8,000 or less, Tucker was the choice of 41 percent, Dixon 18 percent, Clarke 11 percent and 30 percent were undecided.

In households with incomes above $30,000, Dixon won 30 percent, Tucker 25 percent, Clarke 20 percent and 25 percent were undecided.

Citywide, the poll showed no difference in the support for Clarke among tenants and homeowners, despite Clarke's efforts to pitch his campaign toward tenants by citing his record in favor of rent control and antispeculation measures.

Earlier this month, the Metropolitan Washington Council AFL-CIO, the city's largest labor group, endorsed Clarke, saying it was strongly opposed to Dixon's support of measures that labor leaders contend severely reduced unemployment and workers compensation benefits.

Yet the survey found that among union members, who made up 12 percent of those interviewd, Clarke was the choice of 20 percent, while 33 percent were for Tucker, 27 percent for Dixon and 20 percent undecided.

About 14 percent of those interviewed said they were in households where someone was an employe of the D.C. government. Among those respondents, Tucker received 35 percent, Dixon 30 percent, Clarke 14 percent and 21 percent were undecided.

Among residents aged 44 to 60, a segment of the population considered to be among the most dependable voters, Tucker leads Dixon by 37 percent to 25, with Clarke receiving 17 percent and the remainder undecided.

The largest number of undecided voters, 32 percent, was found among those 18 to 29 years of age. Thirty percent of those in this age category said they either support or are leaning to Dixon, compared with 27 percent for Tucker and 11 percent for Clarke.

Among voters 30 to 44 years old, Dixon received 29 percent, Tucker 28 percent and Clarke 20 percent. Tucker was a slim favorite over Dixon among those over 61.