The Los Angeles Times reported that the invitation was extended in a letter obtained by the newspaper.
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Typically, incoming mayors retain some staff from the previous administration -- but extending the offer to all staffers is unusual.
A spokesperson for the mayor-elect says she plans to name her chief of staff immediately.
Bass -- a Democrat who was on President-elect Joe Biden's short list for vice president -- overcame more than $100 million in spending by the billionaire Rick Caruso's campaign while arguing that she would be a coalition builder who could heal a troubled city of nearly 4 million.
Karen Bass to declare a state of emergency on LA homelessness on her first day as mayor
The election tested whether voters in the heavily Democratic city were willing to turn away from their liberal tendencies and embrace an approach that would place a strong emphasis on public safety.
Bass will be sworn in on Dec. 12.
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She will take ooffice as the City Council faces a racism scandal that led to the resignation of its former president and calls for the resignation of two more members. More than 40,000 people are homeless, and there is widespread anxiety over crime that has ranged from daytime robberies on city sidewalks to smash-and-grab thefts at luxury stores.
The Associated Press contributed to this report.