One source said he believed the final decision depends on the vetting of the Democratic governor, much like the selection of Eric Holder to be attorney general.
On Thursday, Arizona Sen. John McCain praised Napolitano.
Also, multiple Democratic sources say billionaire Chicago businesswoman Penny Pritzker is Obama's choice for commerce secretary. Pritzker ran his record-breaking fundraising effort, serving as the campaign's finance chair.
The sources say Prtizker would accept the job, which would be formally offered after vetting is complete. But whether the rigors of the Obama vetting process will present a challenge to a businessperson with no record in public office -- and presumably extensive financial holdings -- is still a question.
Obama met last week with Sen. Hillary Clinton to discuss the possibility of her serving as secretary of state. The Obama team is also vetting her husband, former President Bill Clinton, who has made several concessions in moving the process along. Sen. Clinton's response is expected this week.
However, a Democratic official said Wednesday that Senate leadership is considering a new role for Sen. Clinton should she decide to remain in the Senate.
Senate Majority Leader Harry Reid's motivation for considering a new role for Clinton at this time was unclear, although multiple Democratic sources said Clinton made it known to the leadership prior to her meeting with Obama that she wanted a new role.
Sen. Ted Kennedy, D-Massachusetts, announced Tuesday that Clinton would head a working group on health care reform. Her working group will be focused on insurance coverage. The role Reid is considering would be in addition to that working group position.
"Sen. Clinton has told ... Sen. Kennedy, as well as Leader Reid, that she stands ready to help President-elect Obama in any and every way she can to enact comprehensive health care reform, which she has sought for nearly two decades," Clinton spokesman Philippe Reines said.
Also Wednesday, three sources close to the transition said Obama has chosen former Sen. Tom Daschle to be secretary of health and human services, and the former Senate majority leader has indicated he wants the job.
The sources said that Daschle negotiated that he will also serve as the White House health "czar," or point person, so that he will report directly to the incoming president.
By wearing two hats, Daschle ensures that he -- not White House staffers -- will be writing the health care plan that Obama submits to Congress next year.
The sources said the timing of the announcement has not been worked out, but Daschle is likely to join the Obama transition team as the lead adviser on health issues in the next few weeks.
An Obama transition official would not comment.
However, a source close to Tom Daschle told CNN that Linda Daschle recently informed Baker Donelson that she will be leaving the company at the end of the year in order to open her own independent lobbying shop, which will have no health care clients.
Critics question whether Obama's top staff picks so far represent the "change" that he promised during the campaign. iReport.com: Hillary would be 'more of the same'
More than half of the people named to Obama's transition or staff posts have ties to President Clinton's administration.
Holder, who is expected to be named attorney general, served as Clinton's deputy attorney general. Orszag, who will reportedly head the Office of Budget and Management, was Clinton's special assistant at the National Economic Council and served on the Council of Economic Advisers.
The Clinton-heavy team has caused Republicans to question Obama's call for change.
"I think several individuals are very frustrated to think that President-elect Obama may just cut and paste from some of the Democratic operatives from the Clinton administration and put them into his White House," said Leslie Sanchez, a Republican strategist and CNN contributor.
Republicans aren't the only ones who want Obama to branch out. Robert Kuttner, a liberal and author of "Obama's Challenge," says the president-elect should broaden his recruiting efforts.
"It's not as if the only competent people who ever served in government or who are capable of serving in government are veterans of the Clinton administration, so he's got to be careful how many Clintonistas he appoints to top level government posts," Kuttner said. iReport.com: Who should be in Obama's cabinet?
Before Clinton, however, Democrats had not been in the White House since Jimmy Carter, and many of those in Carter's administration are too old to serve again under Obama.
The Obama transition team said in a statement that they are dedicated to building a well-rounded administration.
"President-elect Obama is committed to putting together a competent team that is diverse in many ways, including experience. Serving in high level positions whether in government, in the private sector or in public service is seen as a positive," spokesman Nick Shapiro said.