The Mariners made the moves on Monday before opening a six-game road trip at Boston and Baltimore.
Morse had been on rehab at Triple-A Tacoma while recovering from a quad injury that landed him on the disabled list for more than a month. Morse played in six games at Tacoma and hit .250 with one double and one home run.
Morse first suffered the injury in late May and has played in just 11 games since May 28.
Bay, 34, signed a one-year deal with the Mariners before the season and his versatility was put to good use in the outfield. But his bat slumped of late and was hitting just .204 with six doubles and 11 home runs.