D.C. City Council member Betty Ann Kane said yesterday she is seriously considering abandoning her financially crippled and foundering bid for the Democratic nomination for mayor and instead seeking reelection as an at-large council member.

Kane said the pressure for her to leave the mayor's race began to build last week after former council chairman Sterling Tucker announced he will run for council chairman again instead of seeking the Democratic nomination for the at-large seat now held by Kane. Kane's term expires next year.

With Tucker not in the at-large race, Kane said, many of her supporters fear that school board member Barbara Lett Simmons could win Kane's seat, while Kane could be out of city politics altogether if she is defeated in the mayor's race.

Many of Kane's strongest backers are public school activists who contend that Simmons' presence on the board has hurt the board's reputation, slowed attempts to improve the schools and would be a negative force if transferred to the council. Kane is a former school board member.

"Sterling's decison not to run for my seat prompted a lot of people to become very concerned about Barbara Simmons," Kane said. "Without Sterling's presence in the at-large race, people are asking me to review my intentions. Many, many people have been concerned about losing me from the council and Sterling's decison intensified it."

Simmons could not be reached for comment yesterday.

Kane said she will make a final decision within a week. Friday night she told a reporter there is a 40 percent chance she will drop out, but yesterday she said it isn't possible to attach odds to her thinking.

"I prefer not call it dropping out so much as deferring being mayor for a few more years," she said.

In early polls Kane has finished third, with less than 10 percent of the vote--far behind the two early frontrunners, incumbent Marion Barry (27 percent) and lawyer Patricia Roberts Harris (38 percent). Kane's departure would affect the prospects of the other candidates, especially the frontrunners.

"Her withdrawal would strengthen in the minds of the public the fact that it is a two-person race," said Sharon Pratt Dixon, Harris' campaign director. " . . . This makes it clear if people want a better city government, there is only one alternative to Marion Barry and that is Pat Harris."

Polls done for both Barry and Harris, which show them toe-to-toe in a tough fight for the lead, indicate that most of Kane's supporters would switch to Harris if Kane leaves the race, according to sources who have seen the poll results.

"It's clearly not in the mayor's best interest," said a Barry campaign strategist who asked not to be named. "It Kane's withdrawal could have two effects. First, there are now no other candidates to draw issues and comparisons with Harris. But secondly--and this is the plus for Marion--it could allow people to take a hard look at Harris' skills and attributes. They may not like what they see."

Tucker, 58, elected chairman in 1974, ran unsuccessfully for mayor in 1978, trailing Barry by 1,500 votes in a close three-way primary.

Sources in the Barry reelection campaign said last week that Barry had urged Tucker to run for chairman with the lure of a possible Barry-Tucker ticket.

Such a slate would permit Tucker, whose campaign organization faltered in 1978, to benefit from the Barry organization in return for the help Tucker could offer Barry in attracting older, middle-class black voters who did not support Barry in the 1978 primary.

Tucker and Barry were not available for comment.

If Tucker were to win the chairman's seat, he would replace Arrington Dixon, who many political observers say views Kane as a primary political rival.

Bob Boyd, cochairman of Kane's campaign organization, said many of Kane's supporters feel that she entered the mayor's race because of her regular disagreements with the current members of the council and her failure to be assigned to a committee chairmanship by Dixon.

"Her talents and skills have not been used to the utmost on the council," Boyd said. "If the leadership of the council is different her aversion to going down there again might be dfferent. Sterling would mean different leadership."

Kane aide Sharon Ambrose said the rigors of the campaign and the scant support Kane has attracted have also contributed to the push to have Kane run for reelection instead of mayor.

"She's campaigned very hard," Ambrose said, "and we're having problems with funding. In every campaign there is a time to assess things . . . It's difficult for a candidate who has psyched herself up to go the whole way to call it quits with so many voters still fluid, undecided. But you've got to know when to stop."

Kane said news of her campaign's financial troubles have contributed to the push to have her leave the mayor's race. In the June 10 campaign finance report, Kane, who has raised $159,261, listed only $22,284 cash on hand.

"It's no secret that we've been struggling with money," Kane said, "and it's been all the more apparent because of all the money the sleazy incumbent has raised. People are concerned about losing trustworthy politicians, people want me involved."

Kane said she is asking her fund-raisers to pledge to raise $200,000 in campaign contributions for the remainder of the race. Without those pledges, she said, she could not remain in the race for mayor.

Realtor Mike Brenneman made a major effort Friday to secure pledges to raise the money necessary to continue Kane's campaign, but met with little success by day's end.

"Everyone feels the woman is far and away the best candidate in town to be mayor," he said, "but they don't think she can win. What really bothers me is that they say she can't win because she's white. And I'm getting this from whites . . . they've got this liberal guilt thing about a black being mayor."

"Right now I'd have to say I'm pessimistic about getting the money," Brenneman said. "Unless I hear something different this weekend, I'm going to have to report to her Monday that the money is not there."

Kane said that if she drops out of the race she has no plans to endorse any other candidate in the race for the Democratic nomination. She said she views the contest now as a two-way match between Barry and Harris, with her council colleagues in the race--at-large council member John Ray and Ward 4 council member Charlene Drew Jarvis--having "no chance at all."

Kane's campaign has dragged since the beginning. Her fundraising lagged from the start and she has had to fight the presumption in some quarters, particularly among potential contributors in the business community, that a white candidate cannot be elected mayor in a city that is predominantly black.

Her support has centered among school parents drawn to her during her tenure on the school board and several neighborhood groups angered at the Barry administration's failure to produce the comprehensive plan for city development that Barry promised during the 1978 campaign and his stance on some commercial development projects.