Drogba was a mainstay of Mourinho's successful side during the manager's first spell at Stamford Bridge, and later netted the penalty which secured a shoot-out victory and the European Cup trophy for Chelsea in 2012 with his last kick for the club.
Chelsea are now preparing to tackle the Turkish giants, and Drogba, in the first leg of their last-16 tie.
"It's a strange feeling," Mourinho said. "Before the game (we are) big friends and (I have) big respect for a legend of this club, after the game the same.
"During the game we have a job to do and he has his own nature and by his nature he wants to win, he wants to score. That's normal."
Asked to compare the Drogba of Mourinho's first spell and now, the Chelsea boss said the striker is "the same".
It is not the first time Mourinho has faced his former talisman, having encountered him when the Portuguese's Inter Milan played Chelsea and then last season when Real Madrid took on Drogba's Galatasaray.
Real lost the quarter-final second leg in Istanbul, but held a comfortable first-leg advantage to advance.
Mourinho knows the size of the task which faces his side, some of whom are yet to experience the Champions League knock-out stages after last season's early elimination was followed by the Europa League win.
"They have a good team, very experienced team," said Mourinho, who will also come up against his Inter playmaker Wesley Sneijder.
"They have Champions League winners, they have big players. Sneijder, Drogba. And after that they have very good players (Emmanuel) Eboue, Felipe Melo.
"They are a top team. Last year my (Madrid) team was a team ready to win the Champions League and we were in trouble against them. We won in Madrid but we lost in Istanbul.
"The crowd is difficult, too. The atmosphere is really strong. We have a big difficult match to play."
Mourinho expects the tie to be alive until the closing stages of the second leg at Stamford Bridge.
"It's a tie to go till the end, really till the end," he added.
In a match with numerous sub-plots there is another as Mourinho faces the manager he succeeded at Inter and Manchester City's title-winning boss in Roberto Mancini.
After weeks in which he has attacked Arsenal's Arsene Wenger and Manchester City's Manuel Pellegrini, Mourinho would not be drawn into discussing Mancini.
"You get nothing from me now," Mourinho said.