Welbeck was hurt in Arsenal's 0-0 draw at Chelsea on Sunday and will miss Wednesday's League Cup match (local time) against third tier side Doncaster, Premier League games with West Bromwich Albion and Brighton & Hove Albion, and a Europa League fixture against Bate Borisov.
"Danny has a scan today, we don't know how bad his groin injury is but he will certainly be out until the international break," Wenger told the club's website.
The 26-year-old, who has scored three times for his club this season, is also unlikely to feature for England in their upcoming World Cup qualifiers against Slovenia on October 5 and Lithuania on October 8, though he could return in time for Arsenal's trip to Watford on October 14.