Burr is only in his first term as a North Carolina senator, but he came to the chamber in 2005 with a decade of experience in the U.S. House.
He is competing for the Republican Conference chairmanship, which is considered the GOP's Number 3 leadership position in the Senate. That position would task Burr with helping his party communicate its message.
Republicans will vote in a closed-door and secret-ballot meeting on Thursday. Burr is competing against Senators Kay Bailey Hutchison of Texas and Lamar Alexander of Tennessee.