The Ivorian frontman, returning for a second spell at Stamford Bridge, landed awkwardly after contesting a header and was unable to run off what appeared to be a twisted ankle.
Drogba attempted to shake off the injury but, after coming back on following treatment from the club's medical staff, had to come off the pitch.
He was replaced by Oscar and hobbled straight down the tunnel to receive further treatment. His possible absence would be a huge blow for Jose Mourinho, with only Fernando Torres available as cover to Diego Costa.
Drogba started alongside Torres in the friendly in Budapest as the Blues continue their preseason preparations. The 36-year-old had one shot on target, which was comfortably saved, before limping out of the friendly.
Chelsea faces Real Sociedad in a final friendly ahead of the new season on August 12 but will hope to have Drogba available for the Premier League opener away to Burnley on August 18.