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, was forced to withdraw and now hopes that he might recover in time to take on the newly promoted team.
"I twisted my ankle and I couldn't continue," he told the club's official website.
"In the next few days we are going to do some treatment and I hope everything will be better."
Drogba was replaced by Oscar and hobbled straight down the tunnel to receive further treatment and his possible absence would be a huge blow for Jose Mourinho, with only Fernando Torres available as cover to Diego Costa.
The Ivorian started alongside Torres in the friendly in Budapest as the Blues continue their pre-season preparations. The 36-year-old hit one shot on target, which was comfortably saved, before limping out.
Chelsea face Real Sociedad in a final friendly ahead of the new season on Tuesday before taking on Burnley on August 18.