Nithya Raman, a progressive member of the Los Angeles City Council, pulled ahead of the reality TV star Spencer Pratt in the Los Angeles mayoral primary on Sunday, as a surge in the vote count signaled a potential shift in the race for second place.

Five days after the election deadline, it remained officially undetermined who will face Mayor Karen Bass, the incumbent, in the November contest to decide who will lead the nation’s second most populous city. The Associated Press, which estimates that about 80 percent of the vote has been counted, has not determined a winner.

But late returns have trended heavily in favor of the liberals who make up an overwhelming majority of the city’s electorate.

Ms. Bass, 72, a former Democratic congresswoman who has already advanced to the November runoff, remained in the lead with 34.7 percent of the vote, according to the updated tally released by the Los Angeles County Registrar-Recorder/County Clerk’s office.

Ms. Raman, 44, who has steadily gained ground as the count has progressed, had 27.1percent, enough to overtake Mr. Pratt by a little more than 3,000 votes. He had been running in second place since election night.

A resident of Pacific Palisades who lost his home in the fire and channeled the fury of his neighbors, Mr. Pratt, 42, a Republican, dropped to third place with 26.7 percent of the vote.

A Los Angeles mayoral candidate can win outright by receiving a majority of votes in the primary. Otherwise, the top two finishers regardless of party advance to the general election. But Los Angeles residents vote by mail in large numbers, and ballot counting is a slow process, with leads often shifting as the tally drags on for days or even weeks. Mail-in ballots with an Election Day postmark are accepted by county election officials through June 9.

On Sunday, Ms. Raman said in a statement that “we are encouraged by the latest vote count and remain grateful to the thousands of Angelenos who have powered this campaign.”

Mr. Pratt did not immediately respond to a request for comment. On social media shortly after the update, however, Mr. Pratt suggested, without providing evidence, that fraudulent votes from homeless people might have affected Ms. Raman’s total.

Alex Stack, a campaign spokesman for Ms. Bass, signaled the likely points of contention between the two if Ms. Raman, ultimately takes the second slot on the ballot. Though her support has been eroded by the wildfires and general voter angst in the city, Ms. Bass has largely maintained the support of the center-left establishment in the city while Ms. Raman has been backed by a younger and more progressive base.

“We look forward to winning a contest against an opponent who allows encampments near schools and fights against hiring more cops,” Mr. Stack said in a statement, “yet is M.I.A. on saving Hollywood jobs and fighting back when ICE invades L.A.”