Score, who died in 2008, served as the radio voice of the Indians from 1968-97. He was inducted into the Broadcasters Hall of Fame in 1998.
Hegan has been a part of the Indians' TV and radio broadcasts since 1989. Since 2007, he has worked exclusively on the team's radio broadcast.
Fans can take part in online voting for the award beginning Thursday at 10 a.m. ET at the Hall of Fame's Facebook site, and concluding at 5 p.m. ET on Sept. 30.
In the fall, a 20-member electorate will vote on the final 10-name ballot to determine the 2012 Frick Award winner, who will be honored in Cooperstown during Hall of Fame Weekend 2012, July 20-23.