Paddack has yet to pitch for the Twins this season while recovering from Tommy John surgery on his right elbow.
Paddack, 27, posted a 1-2 record with a 4.03 ERA in five starts last season, his first with Minnesota.
He is 21-21 in his career with a 4.20 ERA in 66 games (65 starts) for the Padres (2019-2021) and Twins, who acquired Paddack from San Diego in April 2022.
Winder, 26, posted a 2-1 record with one save and a 4.15 ERA in 19 games with Minnesota this season.
Moran, 26, went 2-2 with a 5.31 ERA in 43 relief appearances with the Twins this season.