Costa missed the Blues' defeat to Arsenal in the Community Shield on Sunday after complaining of "discomfort" in his hamstring - the latest in a series of muscular problems that have plagued him since his breakthrough season with Atletico Madrid.
And while Mourinho was confident that Costa would train with the rest of the Chelsea squad on Friday evening, he admits he cannot be sure whether the Spain international will be in condition to take the field at Stamford Bridge the following day.
When asked if Costa would train, Mourinho replied: "I don't know - I promise you!
"I think he's going to train with the team. His reaction to that and confidence to play [on Saturday] is another story."
Mourinho has dismissed the possibility of signing another striker this summer and is confident of Chelsea's chances of making a winning start to their Premier League title defence, with Loic Remy or summer arrival Radamel Falcao up front if necessary.
"I have Remy and Falcao," he added. "I go with them like I did in other matches, like I did last season so many times with Remy and Drogba.
"But obviously [Costa] is a top striker, everyone wants top strikers and he’s such an important player for us - not just for the goals he scores but also for the football he produces and the mentality he helps to create in the team."
Costa scored 20 goals in just 26 Premier League appearances last season and Mourinho insists he is willing to endure the 26-year-old's persistent hamstring issues if he continues to win matches when fit.
"I prefer a situation where the player is never injured, never has problems and has top level during 10 months of competition," Mourinho continued. "But there is nobody to blame.
"I can’t blame the player because he works hard, harder than before. I trust completely in the medical department so I have nobody to blame.
"I prefer a top player like him who makes a real difference when he is ready than a player who is available all the time and just a little bit better than myself!"
