Drogba, who spearheaded Mourinho's successful three-year spell at Stamford Bridge, will be reunited with his former boss when his current club Galatasaray meet the Portuguese's Real Madrid in the Champions League on Wednesday.
Chelsea have gone through seven managers since Mourinho left in 2007 with the sense of instability at the West London club reaching a high point this season with the interim appointment of the unpopular Rafa Benitez in November.
Benitez has already announced he will be quitting the club at the end of the campaign and Drogba believes Mourinho would be the ideal replacement.
"I don't think he's got over Chelsea," the 35-year-old striker was quoted as saying by the BBC on Wednesday. "They haven't got over him."
Mourinho won two league titles, two league Cups and an FA Cup during his three years in charge at Chelsea, but the Champions League remained elusive.
He reached two semi-finals but failed to progress on both occasions and Drogba feels this could tempt him to return.
"When he came, he brought success," Drogba said. "We won league titles back to back with some great football and the only thing he didn't win was the Champions League.
"Maybe that is one of the reasons why he might want to come back. It is unfinished business."
One of the main obstacles in Mourinho's path to Champions League success at Chelsea was former Liverpool coach Benitez, who twice out-smarted the Portuguese in the last four.
He now finds himself sitting in the Chelsea dugout, but his Liverpool connections have made him almost universally disliked by Chelsea fans since he replaced terrace favourite Roberto Di Matteo.
"I think for the club it is the best solution to bring Jose back. The fans want him back. He loves Chelsea," added Drogba, who joined Chelsea in 2004 from Olympique Marseille.
"Because you want to rebuild a team and to give them the strength we had a few years ago, he could be the right person.
"He is a winner and is close to the players. He went to Inter and is at Real Madrid, but is still talking about Chelsea, so he loves this club."
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