ON an Indian summer afternoon, Mayor Frank Jordan is pressing the flesh among merchants on fashionable Union Street, when a woman asks what he is doing about city buses. They run late, drivers are rude, and safety is a concern, she says. ''That's why I'm renegotiating the driver union's contract to change the work rules,'' the mayor responds. Taking on labor is part of Mr. Jordan's reelection strategy to push some of the popular themes of the day, such as cutting government and curbing crime. Given the conservative trend that has transformed politics nationwide, such a tactic seems logical. Yet Jordan, a moderate Democrat, is in trouble in his reelection bid this fall - from someone more to the left. Part of the reason is surely the city and the challenger. San Francisco is notoriously liberal. Moreover, out of a field of seven candidates - none Republican - Jordan's principal opponent is Willie Brown, the well-known and wily former speaker of the California Assembly. Yet many analysts see more in Jordan's campaign troubles than local politics. They see the start of a possible urban backlash against the new conservative agenda in Washington and state capitals. This fall's big-city elections will test whether Republicans can extend their base at the local level - and whether more conservative ideas, from mayors of both parties, can be successfully implemented. ''Cities are the last Democratic bastions,'' says William Schneider, a political scientist at the American Enterprise Institute in Washington. ''Suburban state legislatures have been uniformly hostile to cities. Suddenly, Washington is no longer friendly.'' Indeed, the Republican takeover of Congress last November changed the longtime relationship between Washington and big urban centers. Mayors have long depended on federal assistance for everything from transportation funds to aid for the homeless. Now, with a new mood of fiscal constraint in Washington and suburban Republicans more pervasive in state legislatures, mayors are being forced to rethink the way they manage their cities. This may spell eventual difficulties for sitting mayors of all political stripes - including the class of more conservative mayors such as Jordan here and Richard Riordan (R) in Los Angeles. That's because the performance of local government is easy for voters to judge. Expectations of fiscal restraint may clash with the need to plow the streets, light city parks, and round up stray dogs. Over time, argues Eric Schockman, a political scientist at the University of Southern California, conservative remedies to local budget shortfalls may actually hurt those who advocate them. Mayor Neil Goldsmith (R) of Indianapolis has captured national headlines with his ambitious plans to privatize the city's airport and other services. But at what cost? ''The trade off is a loss of civil service jobs,'' and therefore votes, Dr. Schockman says.'' Privatization is an easy sell, but I'm not sure it is the magic bullet.'' Futhermore, there is little evidence to suggest that party affiliation is helping Republican mayors gain favor from the Republican Congress. ''Republican big city mayors are an anomaly,'' Professor Schockman says. ''They were elected with the proviso that they would have access to the new Congress.'' The Jordan-Brown race illustrates the problem city chiefs face in an era of constricting state and federal budgets. The first round of voting is Nov. 7, and most expect the two to face each other in a runoff in December. Defeating a liberal incumbent in a liberal town four years ago, Jordan promised to be a citizen mayor who would cut waste from government and provide safer streets. He balanced four budgets amid cuts in state funding, and managed to keep all public libraries open. He put 200 more police on the beat without raising taxes. And he opened up city contracts to competitive private bids while retaining for city unions the last right to counterbid. But at the end of his first term, residents of San Francisco aren't sure they want him back. He has angered many different factions of San Francisco society. Bus drivers worried about their own safety complain the mayor has only assigned 50 police to monitor crime on 800 vehicles. The Chinese complain his housing policies compromised their desire to have live-in extended families. Gays criticize what they regard as inadequate health care services. ''Everyone wants to cut out waste, but then it becomes a question of whose ox is being gored,'' says Mark DiCamillo, director of the California Poll. ''If gays, blacks, or Asians think they are taking the hit, a more liberal candidate becomes more palatable.'' Enter Willie Brown, the archetypical liberal California Democrat. Working every constituency, he promises to find the money - through the political connections he forged during 31 years in Sacramento, higher taxes if necessary, and government reforms. Homosexuals, unions, and minorities all contribute to his base. Few doubt Brown would have a hard time meeting the expectations he's building during the campaign. Part of the problem may be the changing nature of cities in an era of a move to balance the federal budget. ''People are saving themselves from urban problems by walking,'' Professor Schneider says. As the impact of federal budget cuts hit, he says, ''there will be an accelerated migration out of the cities. Voters are voting with their feet.'' Those who remain, he says, will be poor minorities.