The Rams announced on Thursday that the ex-England international has agreed a three-year deal with the club.
After calling time on his playing career in 2016, Lampard start his first managerial job with the English Championship club.
And Drogba, still plying his trade with United Soccer League side Phoenix Rising, has wished his former colleague well in his new path.
“I might come play for few months just to support my man. All the very best my man, Frank Lampard,” Drogba wrote on Instagram.