The Ivory Coast forward, 36, suffered the injury and was withdrawn after 28 minutes of the match in Hungary.
He said: "I twisted my ankle and I couldn’t continue. In the next few days we are going to do some treatment and I hope everything will be better."
Drogba, who quit Chelsea following their 2012 Champions League triumph, rejoined the club on a one-year deal last month after 18 months in Turkey with Galatasaray.
Chelsea travel to top-flight new boys Burnley for the first Monday Night Football fixture of the season as Jose Mourinho begins his quest to regain the title and improve on last term's third-place finish.
Watch the first MNF fixture of the season, Burnley v Chelsea, on Sky Sports 1 from 7.00pm on Monday, August 18.