Keane, who met with Trump earlier this week at Trump Tower, told ABC News that he was offered the job as secretary of defense but withdrew his name from consideration because of other commitments and personal reasons. (He lost his wife earlier this year).
Trump transition officials would not confirm whether an offer was made but indicated the president-elect had been very interested in Keane, even as he expressed reluctance to accept a cabinet appointment.
Keane told ABC that at the meeting last week he recommended Mattis and retired Gen. David Petraeus to the president-elect as candidates for the defense secretary job.
Trump met with Mattis on Saturday at Trump's estate in Bedminster, New Jersey, after which Trump said the retired general, who formerly headed the U.S. Central Command, was "the real deal."
Today, Trump tweeted that Mattis "is being considered for secretary of defense."
Federal law requires that defense secretary appointees are at least seven years removed from active duty, so an appointment of Mattis would require a waiver from Congress, if he were nominated and confirmed.
Such an action is not without precedent.