"Having worked with Rex for many years, Rex is going to make the decision that is best for our country," Darren Woods said in a "Squawk Box" interview that aired on Thursday. "His personal circumstances or where he was in the past, I have no doubt that is not going to play in his decision making."
Asked if he's spoken recently to Tillerson, Woods said, "He's been pretty busy, and I've been pretty busy."
Since Exxon has business with Russia, Tillerson was questioned extensively during his confirmation hearing about his ties to Moscow, and whether he perceives Russian President Vladimir Putin as a friend or foe of the U.S. in light of intelligence showing Russia tried to meddle in the 2016 presidential election.
"I don't really have a view or a prognosis on the relationship between our countries," Woods said. "We have a mix of businesses in Russia, we have a very successful business on the east coast that is not affected by the sanctions. We continue to invest and manage [there]."
Woods spoke to CNBC on Wednesday at the New York Stock Exchange, where Exxon held its annual meeting with analysts. He said if he needs help navigating his new job as chairman and CEO of the world's largest publicly traded oil producer, he'll seek counsel from resources within the company.
"I have a great board. We have a very experienced, knowledgeable, diverse board all are very supportive. I had a year as a president serving on the board to get to know all of them," Woods said. "Any questions I've got ... they're my first stop."
The 52-year-old Woods was elevated to president of the company in 2016 in preparation for the 64-year-old Tillerson's mandatory retirement at age 65. The succession plan was speeded up late last year when Tillerson became President Donald Trump's pick for secretary of State. Woods formally took the helm on the first of the year.