The 23-year-old midfielder suffered a hairline crack of his left fibula so will miss Sunday's Premier League opener with West Ham United and is also likely to sit out England's Euro 2016 qualifying Group E fixtures with San Marino and Switzerland next month.
Wenger said on Arsenal.com: "I had a bad surprise because it is a hairline crack in his fibula that makes him a few weeks out."
Wilshere missed last Sunday's FA Community Shield defeat of Chelsea, the Premier League champions, after suffering the injury in training the previous day.
Wenger initially said it would be a "matter of days" before Wilshere returned, but it swiftly became apparent he would miss the Hammers clash.
Wenger added: "There is minimal damage apart from the bone damage – there is no damage at all apart from that.
"It was a collision in training and it was all completely accidental."
Wilshere was in outstanding form for England last time out, scoring twice in the 3-2 win over Slovenia.
