Costa, who scored in Chelsea's league victories over Burnley and Leicester City, picked up a muscle injury in training this week.
"We are trying everything for him to play tomorrow," Mourinho told reporters. "He didn't train for the last two days, but today is the last day and we must make a decision to select him or not.
"He has to try a light training session to see if the reaction is positive. If it is we take him to Everton to play or be on the bench; if the reaction is not positive today then obviously we don't select him."
Spain international Costa, who was born in Brazil, joined Chelsea last month for 32 million pounds ($53.1 million) from Atletico Madrid.