FOX News also confirmed that Chicago businesswoman Penny Pritzker has taken herself out of the running to be secretary of Commerce.
Pritzker was his campaign finance chairwoman, and is credited with Obama's breaking all political fundraising records.
Obama transition team spokeswoman Stephanie Cutter called Pritzker a "trusted adviser and valued friend" to the Obama who "played a critical role in his campaign."
"She would be an enormous asset to an Obama administration given her experience in business and economic growth, but has decided that given her family and business commitments, she is not interested in serving at this time," Cutter said in a written statement.
Gov. Napolitano, whose handling of immigration issues brought her praise from fellow governors, was an early supporter and campaigner for Obama's presidential campaign and was reported to be on a short list of people to fill cabinet posts in the new administration.
Napolitano, 50, was re-elected to a second four-year term in 2006 as governor of Arizona, the home state of Sen. John McCain, the Republican presidential nominee in the race against Obama.
The governor's spokeswoman, Jeanine L'Ecuyer, would not comment on the reports.
"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.
Democratic officials said Napolitano has not been officially offered the job but is likely to be named and to accept, pending vetting.
The reports come on the same day it was learned former Senate Majority Leader Tom Daschle will become Health and Human Services secretary pending Senate confirmation.
Daschle accepted the offer, according to two Democratic sources close to Daschle and with intimate knowledge of the decision. The senator from South Dakota has long been an adviser to Obama's campaign and served as a frequent surrogate on the campaign trail and in media interviews.
The appointment has not been announced, but these officials said the job is Daschle's, barring an unforeseen problem as Obama's team reviews the background of the South Dakota Democrat.