Price, a 33-year-old left-hander, was 16-7 with a 3.58 ERA in 30 starts for the Red Sox, who won their fourth World Series title in 15 seasons. He was 6-3 with a 3.38 ERA in 11 starts and five relief appearances in 2017, when he was slowed by left elbow inflammation.
The 33-year-old Venters was 5-2 with three saves and a 3.67 ERA for Tampa Bay and Atlanta, which reacquired him on July 26. He had not appeared in the major leagues since 2012, undergoing Tommy John surgery for the second and third times.
The awards, announced Tuesday, are voted on by beat writers for MLB.com.