Adler was absent towards the end of 2013 due to his injury, which even after a rest over Christmas is showing no signs of healing.
The 28-year-old was forced to return from the northern Germany club's training camp in Abu Dhabi to undergo tests in Munich.
He has been told to rest for three to four weeks, ruling him out for a short stretch beginning with Hamburg's clash against Schalke on January 26.
"It's obviously very bitter for Rene," said Hamburg's director of sport Oliver Kreuzer on the club's website.
"But with Jaroslav Drobny, we have another goalkeeper who has proven his quality time and time again, and we have faith in him."
The injury and consequent lack of action may also jeopardise Adler's World Cup hopes.
After missing out on a place at the 2010 World Cup, where he was set to inherit the Germany number one jersey from Jens Lehmann following his retirement, Adler is under pressure from Roman Weidenfeller, Marc-Andre ter Stegen and Ron-Robert Zieler for selection in Joachim Low's squad.