PHOENIX — Now that she helped get him elected, the question that remains is whether Gov. Janet Napolitano will become part of the Barack Obama administration.

The possibility of the governor joining Obama’s cabinet was fueled further Wednesday when the president-elect named her as a member of his “transition team.’’

Gubernatorial press aide Jeanine L’Ecuyer did nothing to quell speculation that her boss would resign. She brushed aside questions of whether Napolitano has been asked for the kind of personal and background information that would be necessary to screen potential nominees.

“I just cannot get into that discussion,’’ L’Ecuyer said.

And the governor herself, asked before Tuesday’s election about queries from Obama, would say only that he requested information about state financial matters.

The governor has refused to state flatly she will complete the last two years of her four-year term, saying only she is “not looking’’ for another job.

Napolitano has repeatedly been mentioned as a potential attorney general. She would appear to have the necessary credentials, having been the state attorney general for four years before being elected governor, and the U.S. Attorney for Arizona under the Clinton administration before that.

She also could become the next homeland security chief.

Obama himself told reporters in January, when Napolitano endorsed him, that any Democrat would be wise to consider having the governor in the Cabinet.

“She’s just a singular talent out there,’’ Obama said. He said Napolitano has dealt with difficult issues in “a commonsense way that brings people together’’ and has “the kind of tone and temperament I’d want to see in my government.’’

L’Ecuyer said Napolitano’s agreement to help Obama with the transition team will not impair her ability to deal with the state’s $1 billion budget shortfall for the current fiscal year or to propose a spending plan for the upcoming year.

“She multitasks very well,’’ L’Ecuyer said of her boss.

L’Ecuyer also said it is not clear how much time — if any — Napolitano will need to spend either in Barack’s home town of Chicago or in Washington dealing with transition team duties.

“I imagine sometime in the next 24 to 48 hours we may know a little bit more about exactly how this board is going to work,’’ she said.