Sen. Marco Rubio and other GOP senators fired a warning shot this weekend over President-elect Donald Trump’s consideration of ExxonMobil Chairman Rex Tillerson for secretary of state, raising concerns about the global energy titan’s reported Russian ties.

Tillerson is thought to be a favorite for the position of top U.S. diplomat and met again with the president-elect in New York on Saturday, though Trump would not confirm his selection in an interview with “Fox News Sunday.” The president-elect said only that he’s “getting very, very close” to an announcement.

But Rubio, who ran unsuccessfully for the 2016 Republican presidential nomination and later endorsed Trump, posted a thinly veiled warning about Tillerson on Twitter Sunday morning, without mentioning him by name.

Being a "friend of Vladimir" is not an attribute I am hoping for from a #SecretaryOfState - MR — Marco Rubio (@marcorubio) December 11, 2016

Tillerson is reported to have close ties to Russia. Democrats already have seized on the accounts, with the Democratic National Committee preemptively declaring such a choice “another victory for Vladimir Putin.”

Republicans have taken a more measured – but still cautious – approach.

Sen. John McCain, R-Ariz., chairman of the Armed Services Committee, told Fox News on Saturday that Putin is a “thug” and while he doesn’t know the nature of Tillerson’s relationship with the Russian leader, it’s a matter of concern.

Speaking Sunday with CBS News’ “Face the Nation,” McCain said the Senate would give him a “fair hearing” and noted Tillerson’s ties could be “strictly commercial.”

But he reiterated that “it should be a matter of concern.” He voiced concern that Tillerson’s relationship could “color his approach” toward Putin and the Russian threat.

Amid the bipartisan debate, Trump tweeted late Sunday morning: “Whether I choose him or not for ‘State’- Rex Tillerson, the Chairman & CEO of ExxonMobil, is a world class player and dealmaker. Stay tuned!”

The Wall Street Journal reported last week on Tillerson’s ties to Putin and other world leaders.

Under his leadership, Exxon has steadily expanded its Russian business even as rivals faced expropriation and regulatory obstacles.

In 2011, Tillerson announced an expansive relationship with Russia's Rosneft that will spend years and billions of dollars developing technology to explore and produce oil and gas in icy waters in the Russian Arctic. In 2013, Putin bestowed the Order of Friendship on Tillerson.

Trump, though, still has not announced his choice for the State Department job.

Trump told “Fox News Sunday” that he has “others” as candidates for the post, while lauding Tillerson’s record.

“He's much more than a business executive. I mean, he's a world-class player. … He's in charge of an oil company that's pretty much double the size of his next serious 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 -- for the company,” Trump said. “I have tremendous respect for him.”

House Majority Leader Kevin McCarthy, R-Calif., told Fox News’ “Sunday Morning Futures” that Tillerson would be a “smart pick.”

Noting that experts in the State Department would be on hand to assist the next secretary, McCarthy said Tillerson has an understanding of what world leaders are looking for and “knows what buttons kind of push them.”

Speaking on ABC News’ “This Week,” incoming Trump chief of staff Reince Priebus also defended Tillerson as an “incredible businessman and American patriot.”

While noting a “conclusion has not been made,” he stressed that Tillerson is in the business of finding oil around the world, and said “the fact that he actually has a relationship with people like Vladimir Putin and others across the globe is something that [we] shouldn't be ... embarrassed by.”

Asked about tough questions from Republican senators, he said, “We don't have concerns about confirmation.”

The Associated Press contributed to this report.