That also could leave his availability for the Packers' regular-season opener at Seattle on Sept. 4 in doubt.
"He's going to be a couple of weeks," Packers coach Mike McCarthy said Monday. "They were encouraged once they got all the testing done."
Bostick is not expected to need surgery, but McCarthy would not give any other specifics about the injury.
The second-year tight end was injured after he caught a 9-yard pass on the third snap of the Packers' second offensive possession against the Rams. Bostick remained on the field for one more play before taking himself out.
The Packers have been preparing rookie third-round pick Richard Rodgers for the starting job -- he opened each of the first two preseason games with the No. 1 offense -- but Bostick is a big part of their plans, too. He has taken many of the first-team reps in the two-minute drill throughout training camp. He has the big-play ability the Packers lose without Jermichael Finley. Of all the tight ends on the roster, Bostick most resembles Finley in terms of size and athleticism.
Veteran Andrew Quarless replaced Bostick on that series against the Rams and went on to lead the Packers with four catches for 58 yards.
However, when it came time to replace Bostick in practice Monday -- at least in the two-minute period -- second-year pro Jake Stoneburner filled that role.
There were no other new injuries as a result of the game against the Rams.