The source says this was in no way an official announcement, saying Lee wants to be respectful of Sen. Dianne Feinstein.
At 89, Feinstein is the oldest person serving in the Senate right now but she has not officially announced she will call it a career after this term ends. However, it is widely expected that she will not run for another six-year term next year.
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If Feinstein were to retire, it would likely set off a crowded scramble for the high-profile Senate seat in the country's most populous state.
Lee's announcement also comes a day after Rep. Katie Porter announced a 2024 Senate bid for Feinstein's seat.
Other potential contenders include Rep. Adam Schiff, Lt. Gov Eleni Kounalakis, California Attorney General Rob Bonta and U.S. Secretary of Health and Human Services Xavier Becerra, a former longtime member of Congress.
Feinstein has filed 2024 reelection paperwork with the FEC but has faced criticism recently about her fitness for the job. She rejected those suggestions, telling CNN last year that she feels "absolutely" able to serve fully in her position, adding: "I think that's pretty obvious."
CNN contributed to this article.