Bleday, 25, was used sparingly in Miami last season, appearing in 65 games with 204 at-bats. He finished with a .167 batting average but showed some power in limited duty. For a team struggling on offense, Bleday contributed five home runs and 16 RBIs.
Bleday, who played college baseball for the Vanderbilt Commodores, was a 2019 draft choice of the Marlins.
Puk, 27, is a towering presence on the mound, a 6-foot-7 left-hander who posted a 3.12 ERA with 76 strikeouts to go along with four saves last season. He also finished with a 1.15 WHIP across his 62 appearances.
Puk, a 2016 draft pick of the Athletics, will return to the state where he played college baseball, having thrown for the Florida Gators.