Giambi is expected to be out until at least mid-April with a non-displaced rib fracture he suffered on Mar. 7 in a spring training game after being hit by a pitch.
Ramirez will be out until at least mid-May with a left thumb fracture and ligament sprain. He suffered the injury on Mar. 11 when tagging a runner. Surgery is scheduled for Monday in Cleveland.
Giambi, entering his 20th season in the league, hit nine homers and drove in 31 runs in 71 games with the Tribe last season.
Ramirez is in his first season with the Tribe. The right-hander appeared in 18 games with the Phillies last year, going 0-1 with a 7.50 ERA.