(CNSNews.com) - President-elect Donald Trump told "Fox News Sunday" he's "getting very close -- getting very, very close" to naming his choice for secretary of state.
"I have someone in mind that I think will be really fabulous. I think we're going to have one of the great cabinets ever put together."
Press reports on Monday said Trump has settled on Rex Tillerson, the head of ExxonMobil.
Fox News's Chris Wallace asked Trump about a business executive serving as the nation's top diplomat:
"Well, in his case, he's much more than a business executive," Trump responded. "I mean, he's a world class player. He's in charge of, I guess, the largest company in the world. He's in charge of an oil company that's pretty much double the size of his next nearest competitor. It's been a company that's been unbelievably managed.
"And to me, a great advantage is he knows many of the players. And he knows them well. He does massive deals in Russia. He does massive deals for the company, not for himself, but the company."
Trump praised the other people he's been considering for the secretary of state slot, including Sen. Bob Corker (R-Tenn.) and Mitt Romney. "These are all very different types of people. But when you ask me about Rex, I mean, he's a world-class player. There's no question about it."
Trump's choice for White House chief of staff, Reince Priebus, told ABC's "This Week" on Sunday Tillerson is "an incredible businessman and American patriot and somebody that I think has experience across the globe that very few people have.
"I also think we have got other great choices to make," Priebus continued. "The conclusion has not been made. There's no announcement that's coming today, George, as far as Secretary of State is concerned. So, it's a little premature to be claiming that this is a done deal, but certainly Rex Tillerson is an American icon and businessman that I think is a pretty incredible guy."
Host George Stephanopoulos asked Priebus about Tillerson's "close ties" to Russian President Vladimir Putin:
"Well, look, I mean, he's in the business of exploring and finding oil across the globe. And so you have to go where the oil is at, and the fact that he actually has a relationship with people like Vladimir Putin and others across the globe is something that shouldn't be -- we shouldn't be embarrassed by it, it's something that I think could be a huge advantage to the United States.
"Look, it doesn't mean we're not going to be tough across the world, it doesn't mean that we're not going to always put America first in everything we do. But we have big problems in this world. And I don't think we can solve these big problems by making believe that people don't exist. They exist. We have to talk to people.
"(President Ronald) Reagan talked to people across the world and he was very successful in communicating and treating people with dignity and respect, but putting America first in every single we do," Priebus said.
Sen. John McCain (R-Ariz.), head of the Armed Services Committee, told CBS's "Face the Nation" he has concerns about Tillerson's "close personald relationship" with Putin.
"And, obviously, they have done enormous deals together that that would color his approach to Vladimir Putin and the Russian threat. But that is a matter of concern. We will give him his chance. That's what the confirmation process, that is what advise and consent is all about."
McCain said every one of Trump's nominees deseves a "fair hearing," and Tillerson is no different.