The death of Senator Dianne Feinstein has put Gov. Gavin Newsom of California under immediate pressure to name a replacement, especially given the narrow advantage that Democrats have in the Senate.

Under California law, the governor has the authority to name a successor. Governor Newsom, a Democrat, does not face a deadline, but he is unlikely to leave his party and state without a senator for long as a divided Congress tries to avoid a protracted government shutdown.

Based on an earlier promise, Governor Newsom is expected to name a Black woman to serve in the Senate. He clarified this month, however, that he would only appoint someone on a temporary basis until voters had a chance to weigh in next year — and that whomever he appoints could not be a candidate in the 2024 election for a full term.

That means Governor Newsom will not name Representative Barbara Lee, a Democrat from Oakland and a Black woman, who is among several candidates running in the 2024 race.

In an interview with NBC News more than two weeks ago, back when a vacancy remained a hypothetical possibility, he said that he did not want to give one candidate the advantage of incumbency.

“It would be completely unfair to the Democrats that have worked their tail off,” Governor Newsom told NBC News. “That primary is just a matter of months away. I don’t want to tip the balance of that.”

That has disappointed some Black leaders, who have felt that Governor Newsom was not fully living up to his pledge from 2021. Back then, he tried to assure Black organizations and women’s groups that he would try to restore the racial and gender representation that was lost when Kamala Harris departed the Senate to become vice president.

Naming only a short-term replacement for Senator Feinstein limits the pool of appointees. Some of the most qualified leaders are serving in offices that they are unlikely to abandon for a brief stint in the Senate. Mayor Karen Bass of Los Angeles has been mentioned as a possibility, but she has made it clear that she would not be interested. The same is true of Mayor London Breed of San Francisco.

The names of several other Black women have been floated since the spring, when Senator Feinstein announced that she would not run for re-election. They include Shirley Weber, the California secretary of state who runs elections; Holly Mitchell, a Los Angeles County supervisor and former state lawmaker who is running for re-election; and Angela Glover Blackwell, a civil rights attorney in Oakland and the founder of a research and advocacy nonprofit, PolicyLink.

Ms. Weber, a former state legislator, said on Friday morning that she knew her name had been under discussion. But she said she had not spoken to anyone in a position of authority about the possibility, and she doubted she would want to serve in a time-limited office.

“I never take anything in which I don’t think I can make a difference, and being a senator for a year, I probably couldn’t accomplish an awful lot,” she said. “I do know that all the African American women in the state would love to have an African American senator, not just for a year but for a full term.”

Governor Newsom previously appointed Alex Padilla to the Senate in January of 2021 when Senator Harris resigned to become vice president. Senator Padilla was elected in 2022 to serve first full six-year term.