The Ivory Coast striker was forced to withdraw from a charity match after suffering the problem and leaves the Premier League leaders with no fit strikers for their trip to the Emirates Stadium.
The 37-year-old has started both of the Blues' last two games against QPR and Manchester United after injuries to Diego Costa and Loic Remy.
Drogba, a United Nations Development Programme Goodwill Ambassador, was meant to feature in the Match Against Poverty in St Etienne today but pulled out with the ankle problem.
He wrote on Twitter: "Disappointed not to be able to play the MAP2015 because of an ankle injury but still supporting #EbolaResponse."
Chelsea are currently 10 points clear of Arsenal and should they triumph at the Emirates, a win over Leicester next Wednesday would wrap up the title.
The trip to North London could come too soon for Costa following a hamstring injury suffered against Stoke earlier this month.
The Spain international will return to training this week but manager Jose Mourinho expects the Brazilian-born star to be available for his side's final four games.
"Our target was for Diego to play the last four matches and Arsenal is the sixth match from the end," Mourinho said last week.
"He starts training with rest of the squad next week."
Remy could have more of a chance of featuring after a calf problem with Mourinho claiming the French striker had a better than 50 per cent chance of facing United although he was eventually ruled out.
