CNN's Jake Tapper asked Priebus what job would be "best suited" for him beyond RNC chairman.
"I don't know Jake," Priebus said on CNN's "The Lead." "You're clever to ask the question that way through the side door instead of the front door. But, look it's really not an issue. Honestly, it just isn't. Right now, basically most of us have had two hours of sleep in four days. I think people are taking a breath, getting some of the details figured out. There's just these transition teams, as you know, have been working for months and months. So slowly but surely those details will come out."
Priebus praised Trump's victory speech early Wednesday morning, when he said he'd work for all Americans as president.
"If you look at Donald Trump's first speech, after knowing that he was going to be President-elect of the United States, you saw 2:30 in the morning, this is a person who gave a conciliatory, measured, very appropriate, I think, extremely impressive speech about coming together, building bridges," Priebus said. "There was no bragging, there was no relitigating issues from the campaign trail. And he did that."
He continued: "I'm certain that what's in his heart is to continue reminding America that he wants to be a president for anyone in this country no matter who you are no matter where you are."
And when asked how much of a mandate Trump has, Priebus responded the Democrats are completely out of power and that Republicans will "execute a plan and an agenda and that plan is going to be the one we campaigned on."
He added that Republicans will act on their campaign promises, not specifying if that included Trump's more controversial campaign platform points, such as his Muslim ban.
Earlier Thursday during his Washington trip, Trump was asked by reporters directly about his controversial Muslim ban as he walked away from reporters.
Asked later Thursday by CNN's Erin Burnett about the Muslim ban, Priebus defended Trump.
"He said repeatedly there's no religious test. For you all to be coming back and relitigating something that is now five months old is what the problem is in our country," Priebus said on "OutFront." "The problem is that we have to fill 24 hours a day, seven days a week of cable stations that create these issues that don't exist and turn people against each other. If the media is so interested in the media coming together then they ought to do their job and stop stirring the pot where they don't belong."