The Post said Democratic sources told it late Wednesday Napolitano is Obama's pick because of her expertise in dealing with immigration and national security issues in her state.
Napolitano, 50, who is mid-way through her second term as governor, is on Obama's transition team.
The Post's sources said Napolitano's selection, first reported by CNN, won't be official until she clears vetting.
Gubernatorial spokeswoman Jeanine L'Ecuyer said her boss had nothing to say about the possible appointment.
"She's not looking for another job, and the president-elect will announce his Cabinet selection when he's ready to do that," L'Ecuyer said.