As of now, Cleveland investment banker Mike Gibbons is the front-runner for the GOP primary this May. The filing deadline for the race is Feb. 7.
RELATED: Republican Ohio treasurer drops out of U.S. Senate race
According to Politico, if Vance enters the race, McConnell has said he would prioritize this U.S. Senate race.
Politico is also reporting that Sen. Cory Gardner, R-Colorado, who is the chairman of the National Republican Senatorial Committee, is trying to court U.S. Rep. Jim Renacci, R-Ohio, to run for U.S. Senate.
Brown is one of 10 Senate Democrats up for re-election this November. Republicans hold a narrow lead in the U.S. Senate. There are 51 Republican senators to 49 Democratic senators.