Costa struck twice in Chelsea's final pre-season friendly as they won 2-0 over Real Sociedad at Stamford Bridge on Tuesday, with Mourinho claiming the Spanish striker is ready-made for the Premier League.
But Mourinho sounded a warning for promoted club Burnley ahead of their first game of the Premier League season, claiming Drogba could also be involved on Monday despite having only injured his ankle on Sunday against Ferencvaros.
"He's good," Mourinho said after the win over Sociedad.
"I don't want to say ready for Monday but with the quality of our medical department and the mentality he has, I'm not saying it's impossible for him to recover for Monday."
Drogba has returned to Stamford Bridge to again play under his former mentor after two seasons away from England.
But the Ivorian veteran is still likely to play second fiddle to Costa throughout the campaign with Chelsea's boss indicating the Spaniard looks perfectly suited to his tactics, as well as the football of England's top tier.
Costa signed for Chelsea in July with the London-based club paying a £32million buy-out clause to secure the Brazil-born forward from Atletico Madrid - a price Mourinho still considers a bargain.
"What can we expect? I expect Diego and Diego is a team player who works hard, he's not just about goals," the Portuguese manager said.
"It's about his defensive work, mentality and ambition. This is the player we bought and we were waiting for since last season. This is why we didn't buy a striker in the winter market. We were waiting for him, we have him and hopefully everything goes well.
"He looks adapted to the team, and the Premier League, not just Chelsea, should be happy. Sometimes we lose important players going to other countries and sometimes we bring new top players here.
"You can consider that [a bargain] because the market is crazy and the values are very high. For a striker like him, a champion in Spain, a Champions League finalist, a 25-year-old, we consider the price was very much in our favour."
