The race to fill Vice President-elect Kamala Harris' open Senate seat is underway, but California Gov. Gavin Newsom has so far remained mum about potential contenders.

“No timeline has been established. The process is just beginning to unfold,” Newsom, a Democrat, said during a Monday press conference. “We are working through the cattle call of considerations related to what’s the profile, the right choice to replace Sen. Harris.”

California law allows the governor to appoint a replacement to serve out the remainder of Harris' term once she officially resigns to become vice president. Speculation has been swirling for months over whom Newsom will select to replace Harris after President-elect Joe Biden tapped her as his running mate in mid-August. 

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Asked by a reporter earlier this year whether would-be candidates had approached him about the opening, Newsom joked: "You may be the only one who hasn't - unless you just did."

Harris, who was first elected to the Senate in 2016, would be up for reelection in 2022. Her replacement would serve until then.

Two leading contenders to replace Harris are reportedly Attorney General Xavier Becerra and Secretary of State Alex Padilla, both Democrats. Other potential candidates include Reps. Karen Bass and Barbara Lee, longtime members of Congress, as well as almost every one of California's executive officials, including Lt. Gov. Eleni Kounalakis, state Controller Betty Yee and Treasurer Fiona Ma, according to The Hill, citing sources close to Newsom.

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“'When' is one part of that decision-making process,” Newsom said Monday. "‘Who’ is perhaps the more challenging part of that decision-making process.”

California has never had a Latino representing it in the U.S. Senate, even though Latinos account for nearly half -- about 40% -- of the state's population. People close to Newsom have suggested that diversity is important in selecting a new senator for the state.

"Diversity is a given," Newsom's longtime adviser Nathan Ballard recently told NPR. "It's not going to be someone who looks like Gavin."

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Democracy for America, a progressive political action committee headquartered in Vermont, has called on Newsom to tap a Black woman to replace Harris, the second Black woman to serve in the U.S. Senate and the first from California. But Latino Victory Fund, another political action committee, has pressed Newsom to back either Padilla or Becerra, both Latinos.

“This is an opportunity to increase Latino representation in the U.S. Senate and to break a barrier for Latino elected officials in California,” said Nathalie Rayes, head of the Latino Victory Fund. “We need to break this barrier if we truly want a government that reflects the communities it serves.”