Slowey was dealt by Minnesota to Colorado last month and quickly agreed to a one-year contract with the Rockies to avoid arbitration.
The 27-year-old went 0-8 with a 6.67 ERA for the Twins last season and missed significant time with an abdominal strain. He won 13 games in 2010.
Slowey has compiled a 39-29 record with a 4.66 ERA in 100 career contests (90 starts) for Minnesota.
Putnam, 24, made his major league debut for the Indians in 2011, going 1-1 with a 6.14 ERA in eight relief appearances.