"I've known Rex for many years both in his role as the chairman and CEO of Exxon and as a personal friend. He has the vast experience, ability, and judgment to deal with the very dangerous world we find confronting us. His extensive knowledge of the global situation will be an asset in representing our nation," Cheney said.
"As the chairman and CEO of Exxon, he has operated in nations all over the globe, and managed one of our largest corporations. I'm confident that he will do a superb job promoting our national interests in dealing with the complex and difficult choices that are on the agenda for the next administration."
Rumsfeld, who was President George W. Bush's longtime defense secretary, praised Tillerson's selection in a tweet.
Baker, who was secretary of state under President George H.W. Bush and was White House chief of staff and treasury secretary under President Ronald Reagan, called Tillerson "a friend of mine" and "an excellent choice to head the State Department."
He said he'd told Trump's transition team Monday that he supported Tillerson when the transition team reached out to him about the possibility of tapping the oil man.
Baker said in a statement that Tillerson "has an opportunity to be a very effective secretary of state."
"As CEO of one of the world's largest and best-run companies, has demonstrated the management and negotiating skills, and has the international experience, that are required in the job," Baker said. "Further, I am confident he will understand the global challenges facing the United States and has the intellect to address them."
Baker's firm, Baker Botts, has extensive ties to ExxonMobil, where Tillerson is CEO.
Baker Botts has represented Exxon on cases involving allegations of natural-gas price fixing
, issues related to gas pipeline
rates and others. Baker's firm has also represented Russian state-owned oil company Rosneft on a deal with Exxon
Bush also praised the pick Tuesday, calling Tillerson someone with "vast experience" and a "patriot." Early in the race, Tillerson supported Bush's run for president in the 2016 election.
"My guess is what they'll find is a guy with vast experience," Bush told WJHL in Johnson City, Tennessee
. "A patriot, someone who will represent the US really, very well. But they'll have a chance to discuss this is at an opening hearing. That's how the process works."