Greene, with his seventh team in nine years, was thought to be protection for Josh Bard and Rob Bowen. The backstop hit .289 in 61 games for San Francisco in 2006, with a .428 slugging percentage. He batted .326 with two homers and 16 RBIs in 37 starts behind the plate.
The timetable for his return is unclear but he could miss up to six months if the MRI reveals a tear.
In other news, the Padres worked out former farmhand Rusty Tucker on Monday. The left-hander missed most of the 2004 season after undergoing Tommy John surgery.
Once armed with a fastball that would sit in the mid-90s and hit 98 MPH, landing him on the Padres' 40-man roster, Tucker suffered bouts of extreme wildness upon his return to the mound and was eventually removed from the 40-man roster to make room for Mike Piazza in 2006. The Chicago White Sox claimed him, but he is out of work now.
Tucker pitched in front of four members of the Padres staff, including Grady Fuson, Bill Bryk, Mike Couchee and Gary Lance.
He sat 88-91 MPH with his fastball, was down in the zone and showed an average to above average curveball.
The Padres currently don't have much room on the farm but could keep him in mind if a pitcher is injured.