The 42 year-old left-hander underwent an MRI examination Friday morning, when the extent of his arm pain was discovered.
Glavine had avoided the 15-day disabled list during his first 21 major league seasons. When he was placed on the DL Tuesday, it was for the second time in less than two months.
Atlanta General Manager Frank Wren said the injury cannot be repaired through surgery, only rest.
"Sometime around the All-Star break or a little thereafter," Wren said Friday during a conference call with reporters when asked about a specific date for Glavine's return.
The 305-game winner is 2-3 with a 4.85 ERA in 12 starts (59 1/3 innings) this season, but has won only twice in his last 16 starts, dating to last season.
Members of Congress to keep receiving porn magazine
Disney's 'Jessie' to feature network's first engagement