The Milan manager is looking to bolster his attacking options after a disappointing start to the Serie A campaign, which sees the Rossoneri in sixth place.
The 34-year-old ended a decorated eight-year spell at Chelsea on a high by scoring the winning penalty in the Champions League final in May before moving to Shanghai Shenhua in the summer.
The Ivory Coast international is reportedly unhappy in China and wants to terminate his contract, alerting some of the top clubs in Europe.
With Serie A champions Juventus also interested in Drogba, Allergri would relish the chance to work with the Blues legend to improve his squad.
“Champions of his [Drogba] calibre are very welcome. [Vice-president Adriano] Galliani will know if the deal is feasible,” Allegri told Gazzetta dello Sport.
“I’m calm, the club is just thinking about what could help us out and what we could do about it.
“A player like Drogba would certainly improve both the technical level and self-esteem of the team.
“We’ll see what the club are able to do, otherwise we’ll carry on as we are, looking to have a good second half of the season because that is our objective.
“It will be difficult, though, because we still have to face some teams with a high level of technical quality.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news