A's Manager Bob Geren told reporters in Arlington, Texas, before a game against the Texas Rangers the right-hander will require elbow surgery.
"He's going to be missed, both on the field and off," Geren told the Oakland Tribune. "He's been a great teammate and a pleasant guy to be around."
Sheets, who signed a 1-year, $10 million deal with the A's before the season, missed the entire 2009 campaign after tearing the same flexor tendon.
Sheets, 32, had compiled 4-9 with a 4.53 ERA in 20 starts for Oakland this year. He spent the first eight years of his career with the Milwaukee Brewers and holds a 90-92 career record with 3.79 ERA.