Russian President Vladimir Putin and ExxonMobil chief executive Rex Tillerson shake hands in 2012 at a signing ceremony of an agreement between state-controlled Russian oil company Rosneft and ExxonMobil in Tuapse, in southern Russia. (Mikhail Klimentyev/AP)

At the start of the week, Sen. Marco Rubio (R-Fla.) seemed bent on opposing the nomination of Rex Tillerson for secretary of state.

By the end of it, Rubio had heard directly from former vice president Richard B. Cheney, according to a person with knowledge of the conversation, as well as other key supporters of the ExxonMobil chief executive.

With that, one more controversial proposal by President-elect Donald Trump at risk of running aground had found new hope for smoother waters. In this case, the intervention of a high-powered gaggle of establishment Republican leaders — none of them long-standing allies of Trump’s — appears to be leading the effort to save Tillerson’s nomination.

Tillerson is opposed by many Republicans and Democrats who are suspicious of his perceived close ties to Russian President Vladi­mir Putin, who has worked with ExxonMobil on energy deals.

Yet he is also backed by a small army of hawkish GOP luminaries and deep-pocketed donors. Robert M. Gates, a former CIA director and former defense secretary, personally recommended Tillerson in a meeting with Trump. Condoleezza Rice, the former secretary of state, made the same recommendation to Vice President-elect Mike Pence.

(Sarah Parnass/The Washington Post)

Several of these Tillerson boosters enjoy direct lines of communication to Rubio and other Tillerson skeptics such as Sens. John McCain (R-Ariz.) and Lindsey O. Graham (R-S.C.). They are seeking to use those channels to blunt hostility in the Senate, where Tillerson must win confirmation.

[Trump wasn’t happy with his State Department finalists. Then he heard a new name.]

What is not clear is whether these Republicans are working with Trump’s blessing or cooperation — or whether, instead, they are freelancing their own agenda to promote an official they favor for Trump’s administration. Rice and Gates run a consulting firm that counts ExxonMobil as a client.

Some emissaries say they are acting on their own while others say they have been nudged to reach out by those shepherding Tillerson’s nomination. A Trump transition spokesman did not respond to requests for comment.

Rubio, meanwhile, is fresh off a career-saving reelection to his Senate seat and, according to associates, eager to establish himself as a leading voice on foreign policy in the next Congress. He sits on the Senate Committee on Foreign Relations, the first hurdle Tillerson must clear on Capitol Hill.

The Floridian’s navigation of the Tillerson confirmation process early next year will offer one of the first concrete signs of how he plans to position himself on world affairs during the Trump presidency. It will also serve as a test of how effective the kind of behind-the-scenes nudging underway from Republican power brokers will be in the era of Trump.

Republicans hold a 10-to-nine advantage on the foreign relations panel. If just one Republican and all Democrats band together against him, they could sink Tillerson’s nomination before it even reaches the Senate floor.

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Sen. Bob Corker (R-Tenn.), the committee’s chairman, wrote on Twitter on Friday that he had a “very good conversation” with Tillerson on Thursday. Corker said he also received a call from former president George W. Bush, who vouched for Tillerson. The alignment of Bush and Trump on Tillerson comes after Trump, as a candidate, repeatedly criticized Bush’s decision to invade Iraq.

Tillerson has the support of Senate Majority Leader Mitch McConnell (R-Ky.). Many of his boosters are confident that over time he will sway enough Republican senators to back him. But his supporters are leaving little to chance.

Gates has said that Tillerson’s experience doing business with Russia is an asset. He is a “tough-minded realist” who understands Putin, he said. And Cheney has been an outspoken supporter of Tillerson, calling him an “inspired choice” in a statement. Like Tillerson, the former vice president has deep roots in the energy industry, having served as chief executive of Halliburton.

Earlier this week, Cheney spoke to Rubio, said the person with knowledge of the exchange, who spoke on the condition of anonymity to describe the private chat. He is making calls to multiple senators, said a second person close to GOP leaders who spoke on the condition of anonymity to describe private conversations.

The former vice president could not be reached for comment. Rubio declined to be interviewed. A Rubio spokesman did not immediately comment on the conversation.

Robert McNair, a Texas-based GOP donor who gave $500,000 to a super PAC supporting Rubio for president, said he plans to call the senator from Florida to coax him to back Tillerson. He said he also intends to talk to McCain and Graham.

McCain has been one of the leading skeptics of Tillerson’s nomination, questioning his relationship with Putin and saying on CNN: “I don’t see how anybody could be a friend of this old-time KGB agent.”

During his presidential run, Rubio labeled Putin a “thug” and a “gangster” — rhetoric that clashed sharply with Trump’s much warmer statements about wanting to improve relations with Russia.

On Sunday, Rubio attracted widespread attention with a tweet declaring, “Being a ‘friend of Vladimir’ is not an attribute I am hoping for from a #SecretaryOfState.”

Leaving no uncertainty about whether a staffer might have written it, he signed the tweet “MR.” The message has been retweeted nearly 9,500 times.

The worldview Rubio underscored during his presidential run, which was a continuation of a posture he had taken in the Senate, was by and large in line with traditional Republican hawks, including McCain and Graham, who have tended to be wary of Russia’s intentions.

What remains unclear is whether Tillerson’s solid support among many credentialed Republican establishment figures will sway these hawks.

[Some Republicans don’t like Rex Tillerson. But blocking him would be historically rare.]

After Trump officially tapped Tillerson on Tuesday, Rubio released a statement saying that he has “serious concerns” about his nomination but will work to “ensure he receives a full and fair but also thorough hearing before the Senate Foreign Relations Committee.”

Tillerson is facing criticism coming from people who don’t “have any business experience and don’t know how the world works,” said McNair, the owner of the Houston Texans football team. He started an energy company in the 1980s and said he did “a lot of business” with ExxonMobil back then.

Roy Bailey, a Texas-based insurance executive who also gave to Rubio’s presidential campaign, said he backs Tillerson. The resistance to Tillerson, he said, “sounds like old politics to me as opposed to a new way of thinking.”

Tillerson won Russia’s Order of Friendship award in 2013, and ExxonMobil has an active presence in Russia. Anything Russia-related is under extra scrutiny in the political process these days, following a CIA assessment that Russia meddled in the election to try to help Trump win.

One Republican wild card is Sen. Rand Paul (Ky.), who also sits on the Foreign Relations Committee. His libertarian-leaning views have often collided with party leaders. In an interview with CNN earlier this week, Paul said he has “an open mind” on Tillerson.

For the first time in years, Rubio is not gearing up for or making a run for president. And he is fresh off securing another six years in the Senate with his win in November — which came after a damaging presidential campaign that he folded after getting blown out in his home-state primary. Now, those familiar with his plans say, he wants to focus heavily on international affairs from his position in the Senate.

“Anyone who knows Marco well, and I’m one who does, knows foreign policy is his first, second and third interest,” said George Seay, a Texas-based Rubio donor. “I think Marco will plant his flag in that area.”

For now, Rubio isn’t showing his hand when it comes to Tillerson. Associates say they are convinced he is keeping his options open.

“I don’t think he’s made up his mind,” Seay said. “I think he’s just skeptical and concerned.”

Paul Kane contributed to this report.