Wilshere sustained the injury during an accidental collision in training on Saturday and Wenger initially announced that the problem would only rule the England international out for a matter of days.
But the 23-year-old, who sat out Sunday’s 1-0 Community Shield victory over Chelsea at Wembley, is now facing a longer spell on the sidelines and is also a doubt for England’s European Qualifiers with San Marino and Switzerland in September.
“The information you have heard about Jack Wilshere is right,” Wenger told Arsenal’s website.
“I had a bad surprise because it is a hairline crack in his fibula that makes him a few weeks out.
“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.”
The club have also confirmed that Wilshere will continue to be assessed by their medical team as they attempt to fast-track him back to full fitness.
Wilshere has been plagued by a long line of injury problems during his early career and missed five months of last season after sustaining ankle ligament damage in a defeat to Manchester United in November.
This is the fifth year running Wilshere’s campaign has been disrupted by a spell in the treatment room.
Wilshere spent 14 months on the sidelines in 2011 after sustaining a stress fracture of his right ankle in a pre-season friendly with New York Red Bulls.
The severity of the injury also saw him miss the beginning of the 2012/13 season, while a hairline fracture of his left foot ruled him out for six weeks of the following campaign.
A challenge from United defender Paddy McNair led to another season interrupted by injury for Wilshere, who returned only for the remaining six matches of the 2014/15 campaign.