The 26-year-old was taken off 73 minutes into the Gunners' goalless draw with Chelsea at Stamford Bridge on Sunday afternoon after pulling up.
Wenger's initial fears have now been realised as the England international is expected to be out of action for around a month, ruling him out of games against West Bromwich Albion and Brighton & Hove Albion in the Premier League, as well as Doncaster Rovers in the EFL Cup and Bate Borisov in the Europa League.
"Danny has a scan today," Wenger told reporters on Tuesday afternoon. "We don't know how bad his groin injury is but he will certainly be out until the international break."
Arsenal return to action on October 14 following the second international break of the season, with a trip to face Watford at Vicarage Road.