The 31-year-old was expected to miss the Gunners' first game of the season – at home to Sunderland on August 18.
However, Prague daily newspaper Sport reports that the latest procedure could rule him out of action for a further six to eight weeks.
That could mean Rosicky misses as many as six Premier League games, including trips to Liverpool and Manchester City as well as a London derby at home against Chelsea.
"There was some swelling that we needed to get rid of," Sport quoted Rosicky's agent Pavel Paska as saying. "Tomas has already started physio."
Rosicky picked up the injury during Czech Republic's second group match against Greece, and he played no further part in the tournament as his country reached the quarter-finals.
Arsenal will already be without long-term absentee Jack Wilshere and right-back Bacary Sagna for the start of the new campaign as they both battle to overcome their respective injuries.