The team also announced Wednesday that it had agreed to a one-year major league deal with infielder Kyle Farmer, who gets $640,000 while in the major leagues and $175,000 while in the minors.
Bradley and Goodwin were both acquired in midseason trades as the Reds made a push for the playoffs. The right-handed Bradley, 28, came over from the Diamondbacks and pitched well in his six regular-season appearances with Cincinnati, giving up just one run over 7⅔ innings. He allowed the deciding run in Game 1 of the team's National League wild-card series against the Braves, which Atlanta won 1-0 in 13 innings.
Goodwin struggled with the Reds, batting just .163 in 49 at-bats. The 30-year-old played the first half of the season with the Los Angeles Angels.
Casali, 32, spent the past three seasons in Cincinnati and hit .224 with six homers in 76 at-bats in 2020.
The 30-year-old Farmer is a versatile player who can play all positions on the infield and is even an option at catcher. He hit .266 last season.
The Reds also declined to offer a contract to right-hander R.J. Alaniz, who pitched in 12 big league games in 2019.