I Bianconeri have already made an approach to sign Athletic Bilbao forward Fernando Llorente on a Bosman free transfer in the summer, but with Arsenal loanee Nicklas Bendtner sidelined through injury, the Italian champions are eager to bolster their immediate options.
Marotta told television programme Domenica Sportiva: "Drogba? We are always making enquiries and we have made one for him. We have asked about his availability and after that we will look at the financial aspects and talk to the coach."
Juve coach Antonio Conte has publicly ruled out any approach for the former Chelsea man, but player recruitment is overseen by Marotta.
Drogba has been training with the Blues, with whom he won the UEFA Champions League in May, in the lead-up to the CAF Africa Cup of Nations, which begins in South Africa on 19 January.