Galatasaray have made a mediocre start to the season but Drogba believes they can repeat April's victory over Real in Tuesday's Champions League Group B opener.
The veteran striker has been named in a 23-man squad for a friendly against Mexico in New York next Wednesday.
Didier Drogba came off the bench to score two late goals in a 2-1 victory against Arsenal for Galatasaray to win the Emirates Cup.
It is believed that Mourinho is keen on signing 35-year-old Drogba for this season, but Chelsea would have to approach his current club Galatasaray.
The Ivory Coast international, now with Turkish team Galatasaray, scored 157 goals in 341 appearances for the Londoners.
Ivory Coast players Eboue and Drogba, who was instrumental in Chelsea's run to the Champions League title last season, were heckled during Sunday's 2-1 league win by Fenerbahce.
Real reached the Champions League semis for the third straight season after Ronaldo scored twice in a 3-2 loss to Galatasaray.
The Turkish club lost its Champions League first-leg quarterfinal 3-0 at Santiago Bernabeu. In another second-leg tie Tuesday, Borussia will host Malaga.
Chelsea have gone through seven managers since Mourinho left in 2007.
The wheel will have come full circle for the Galatasaray forward when he takes to the pitch for Wednesday's Champions League quarter-final first leg match at Real Madrid.
Drogba disappointed coach Sabri Lamouchi after team's quarter-final exit at the African Cup of Nations earlier in the year.
Drogba needs to return to top form but is still very much in Ivory Coast's international plans, coach Sabri Lamouchi said just days after dropping his team's captain.
Galatasaray striker Didier Drogba was a shock omission when Ivory Coast named a 28-man squad for next weekend's World Cup qualifier.
Drogba, who took the winning penalty for Chelsea in last season's final, scored with a header five minutes into his Galatasaray debut on Friday.
Drogba signed a 2 1/2-year contract with Shenhua but left the club well before that.
Drogba can play in a Champions League last-16, first leg at home to Schalke next Wednesday.
Drogba is teaming up with Wesley Sneijder at Galatasaray, giving the Turkish club a second major signing in a week.
The 33-year-old French striker conducted the standard medical tests at Turin's Istituto di Medicina dello Sport centre.
The 34-year-old moved to Shanghai Shenhua after helping Chelsea win the Champions League last May.
Milan's pursuit of Drogba means Serie A leaders Juventus are likely to go for other striker targets including Olympique Lyon's Lisandro Lopez.