- Tillerson needs to be confirmed by the Senate to serve a secretary of state
- Some Senate Republicans are resistant to him serving in the role
Corker is the chairman of the Foreign Relations Committee, which will handle the first confirmation hearings for Tillerson.
Corker, who was a finalist for the secretary of state job himself, tweeted, "Had a very good conversation with Secretary of State nominee Rex Tillerson yesterday and am looking forward to meeting with him in January."
He added, "Also appreciated a call I received from President George W. Bush, who spoke highly of Mr. Tillerson and offered support for his nomination."
Tillerson will have to be confirmed by the Senate to serve as head of the State Department -- but has met resistance from Senate Republicans for the leadership role.
Republican Sen. John McCain said in a radio interview
Tuesday that, unlike Tillerson, he wouldn't have accepted an award from Russian President Vladimir Putin.
Putin awarded Tillerson Russia's prestigious Order of Friendship in 2013, one of the highest honors Russia gives to foreign citizens.
McCain is not the only Republican senator who is resisting confirmation of Tillerson to the State Department.
"Being a 'friend of Vladimir' is not an attribute I am hoping for from a #SecretaryOfState," Republican Florida Sen. Marco Rubio said in a tweet signed with his initials on Sunday.
However, not all establishment Republicans
are resisting Tillerson. Trump's nomination of Tillerson drew praise from former Vice President Dick Cheney, former Secretary of State Jim Baker, former Defense Secretary Donald Rumsfeld and former Florida Gov. Jeb Bush.