Rodgers has not played this season after undergoing shoulder surgery in March.
Rodgers and Montes fill roster spots vacated when Randal Grichuk and C.J. Cron were traded to the Los Angeles Angels on Sunday.
Rodgers, 26, earned a Gold Glove at second base last season, and the Rockies had high hopes for their 2015 first-round draft pick (No. 3 overall) heading into this season. Over 137 games in 2022, Rodgers batted .266 with 13 home runs and 63 RBIs.
A shoulder injury in the spring ended up costing Rodgers the first four months of the season instead of the entire campaign as originally believed.
In parts of four seasons with Colorado, Rodgers is a career .266 hitter with 28 home runs and 123 RBIs.
Montes, 26, was batting .184 with one home run and three RBIs in 18 games with the Rockies this season. The rookie made his major league debut June 11.
Cron, 33, was in his third season with Colorado and batting .260 with 11 home runs and 32 RBIs over 56 games. He earned his first All-Star nod in 2022, finishing the season with 29 homers and a career-best 102 RBIs.
Cron returns to the Angels after starting his MLB career there in 2014. He is a career .260 hitter with 186 homers, 193 doubles and 599 RBIs across 10 seasons with the Angels (2014-17), Tampa Bay Rays (2018), Minnesota Twins (2019), Detroit Tigers (2020) and Rockies.
Grichuk, 31, is batting a career-high .308 with eight homers and 27 RBIs over 64 appearances. He has batted .251 over his 10-year career with the St. Louis Cardinals (2014-17), Toronto Blue Jays (2018-21) and Colorado.
The Rockies received right-handed pitcher Jake Madden and left-hander Mason Albright in the deal with the Angels. Both were at Single-A this season.