The decision on Friday came after the commissioner's office failed to approve Thursday's transactions.
Tampa Bay could select Kelley to the roster March 15 or later.
Kittredge is recovering from Tommy John surgery and last pitched on June 7. The right-hander went 3-1 with five saves and a 3.15 ERA in 17 games last year.
Kelley appeared in 18 games with Milwaukee in 2022, going 1-0 with a 6.08 ERA. He also been with Boston and Philadelphia.