Goal reported on Wednesday that the Brazil-born Spaniard had sustained a minor muscular injury in training and would be assessed by Chelsea's medical staff to determine the extent of the damage.
Mourinho insists he remains open-minded to the possibility of Costa featuring at Goodison Park but revealed his £30 million man has missed two days of training with the injury.
"He is a big doubt, let's see," the Chelsea boss told reporters. "Everyone else is fit.
"We are trying everything for him to play [on Saturday]. He didn't train for the last two days. Today is the last day.
"We have to try and see if the reaction is positive. If the reaction is positive, we take him and select him on the bench."
Costa has made an impressive start to his Chelsea career, scoring two important goals in his first two Premier League matches against Burnley and Leicester City.