Adebayor sustained his injury in a heavy tackle by Potters defender Ryan Shawcross, while winger Theo Walcott (shoulder) and right-back Bacary Sagna (ankle) were also hurt during the encounter.
Captain William Gallas, who has been out since the 4-4 draw with Tottenham Hotspur with a hamstring problem, could, however, return for Saturday's Barclays Premier League match against Manchester United.
"Adebayor will be three weeks," manager Arsene Wenger said. "Walcott could be days or weeks. It is the shoulder he had the surgery on. We don't know about Gallas yet but Sagna will be back in the squad (for the Fenerbahce match on Wednesday)."
Adebayor is now set to miss the league matches against United, Aston Villa and Manchester City. His strike partner Robin van Persie will sit out the United and Villa matches as well as the Carling Cup encounter against Wigan Athletic next week as he serves a three-match suspension after being sent-off at Stoke.
With Eduardo a few weeks away from playing in the first-team after his broken leg, Wenger's attacking options could be reduced to the inexperienced Nicklas Bendtner, Carlos Vela and Abou Diaby.
Blessing in disguise!!!!
alam shah to replace adebayor want?