As United States-Russia relations deteriorated toward the end of President Obama’s first term and into the second, American diplomats in Moscow took to keeping a list of American citizens who maintained red-carpet access to Russian leaders – including President Vladimir Putin – even as their own dried up.

Near the top of the list was Rex Tillerson, the chief executive of ExxonMobil who is now President-elect Donald Trump’s nominee to become secretary of State.

As Mr. Tillerson undergoes two days of Senate Foreign Relations Committee confirmation hearings beginning Wednesday morning, the Texas oilman’s ties to Russia – and in particular the close relationship he appears to have developed with Mr. Putin and some of his key confidants – will be placed under the microscope.

Tillerson has managed to add some nuance to the initial cartoon image of him as best buddies with one of America’s chief geopolitical adversaries as he met privately over recent days with a number of senators, Republican and Democrat. What emerged in those meetings, several senators and Senate aides have said, is a realist who balances an understanding of the challenges Russia poses with recognition of the need to work with Russia on other key interests.

Some Democratic senators say they have not heard enough from Tillerson to change their view that, as a lifelong businessman with no diplomatic experience, he is not fit to be the nation’s chief diplomat. That’s especially true, they add, given he would be serving under a president who likewise has no formal diplomatic experience.

But one pleasant surprise for some Democrats in particular: The man who served a decade atop the world’s largest oil company says he sees climate change as a critical global challenge – unlike the president he would serve under, who has called climate change a “hoax” perpetrated by the Chinese – and he supports the Paris climate accords.

But most senators and foreign policy experts suggest Tillerson will make or break his own case with his testimony. Aside from probing the Russia connection, senators who have never faced a secretary of State nominee with an impressive international business record but no public diplomatic experience will want to know how Tillerson perceives the difference between the two.

“Tillerson has to dispel the notion that he has any kind of illusion about Russia, that somehow he believes all you have to do is make nice and lift sanctions and all will be rosy with Russia,” says Nikolas Gvosdev, an expert on Russia and professor of national security studies at the US Naval War College in Newport, R.I.

“He’s made some progress in that direction, he’s made it clear he’s not naïve about Russia,” Dr. Gvosdev says. “But I think he’ll have to go farther and demonstrate he understands that as secretary of State his motivation for engaging with Russia will be to advance US national security interests,” he adds.

“He needs to show he understands how that’s different from the dealmaking he did with Russia and other countries on behalf of Exxon.”

'Not some pushover'

Based on reports out of meetings Tillerson held with senators, the oilman with four decades of experience with Exxon knows he has some convincing to do.

Tillerson did not seek to revise the notion that he built a business friendship with a hands-on Putin as the two hammered out energy deals. How could he, when just last February he told an audience at the University of Texas in Austin that he had “a very close relationship” with the Russian leader? But he shared his assessment that Putin is no Mr. Nice-Guy. He characterized Putin as a regional bully and told several senators that he came to know a leader who only respects strength.

The other side of that coin is that, according to numerous reports, Putin was intrigued by the Exxon chief. He took to him – eventually awarding him Russia’s “Order of Friendship” in 2013 – because he was a tough and smart negotiator who knew both sides had to come away from the table with something, but stood his ground when he had to.

“Tillerson was respected in Russia; it was known he was not some pushover,” Gvosdev says. “At a number of points” in negotiations over an Arctic oil extraction deal “Tillerson signaled he was prepared to walk away,” he adds, “and apparently Putin took note of that.”

The tough realist and proven dealmaker in Tillerson have already impressed some key senators. Foreign Relations Committee Chairman Bob Corker (R) of Tennessee says his meeting with the nominee convinced him Tillerson is well within “the mainstream” of US foreign policy thinking, and he predicts overwhelming support for his confirmation.

The human rights question

But other senators and forces within the foreign policy community maintain that being secretary of State goes beyond dealmaking to the promotion globally of American values such as democracy and human rights – and they are not sure Tillerson grasps that part of the job.

“Tillerson played a different role formerly. As chief executive of ExxonMobil he was not responsible for promoting human rights. But as secretary of State promoting human rights will be his responsibility,” says Margaret Huang, executive director of Amnesty International USA in Washington.

Noting that Tillerson as Exxon chief worked closely with governments with poor human-rights records, including Russia and Equatorial Guinea, Ms. Huang says, “We simply don’t know whether Rex Tillerson believes human rights are a critical value.”

But she says she expects the Senate hearings to answer such questions. “We’re pressing senators to push Tillerson to clarify whether his willingness to work with repressive regimes reflects just his business practices or his worldview,” Huang says.

Election meddling, and fallout

Tillerson’s business dealings with despots don’t sit well with senators who favor a robust, values-driven foreign policy, but it is his Russia connection that will get the most attention. Those Moscow ties became more problematic Friday when US intelligence chiefs disclosed their conclusion that Russia under Putin’s direction carried out a covert cybercampaign aimed at boosting Trump in the presidential election.

Some senators want Tillerson to commit to supporting new sanctions on Russia over the election interference.

“Mr. Tillerson has got to convince me and I think other members of the [Senate] that he sees Russia as a disruptive force, that he sees Putin as undermining democracy all over the world, not just in our backyard,” Sen. Lindsey Graham (R) of South Carolina said Sunday. “He has to realize that the Russians did it when it came to the hacking and that new sanctions are justified.”

Yet while Senator Graham and other Tillerson critics like Sen. John McCain (R) of Arizona, who blasted the nominee over his “friendship with a butcher,” are leaving room to eventually vote to confirm him, others are not.

“I am looking for a secretary of State with a breadth of diplomatic experience to balance Trump’s lack of experience,” Sen. Chris Murphy (D) of Connecticut said in a statement after meeting with Tillerson Monday evening. Adding that he also wants a chief diplomat who will “be a voice for protecting international human rights” and will “lead progress on climate change,” Mr. Murphy concluded Tillerson does not meet those standards “and I will continue to urge my colleagues to oppose his nomination.”

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The Navy War College’s Gvosdev says Tillerson the CEO and realist was always going to have a hard time winning over the more ideological and values-driven quarters of the Senate. But he says the nominee should have an easier time convincing senators that he understands the difference between running a corporation and being the nation’s top diplomat.

“There’s little doubt that US diplomacy under Tillerson would become more transactional and deal-driven in a way it hasn’t been for many years,” Gvosdev says. “But I think he can argue that he’ll know how to use that ability to get the best outcomes for the United States and its interests.”