Costa has not played since limping off just 10 minutes after coming on as a substitute against Stoke City on April 4, with a recurrence of a hamstring injury.
He will again watch from the sidelines as Chelsea welcome Manchester United to Stamford Bridge on Saturday, but could be involved at the Emirates Stadium in nine days' time.
"[Costa] is progressing very well. I think he has a chance for the next weekend," Mourinho said.
"Our target was for Diego to play in the last four matches. Arsenal is the sixth match to the end so if he doesn't play [against] Arsenal or Leicester [the following Wednesday], it is not a problem.
"He starts training with the rest of the squad next week."
Despite being limited to 24 appearances through injury, Costa is the Premier League's joint top-scorer with 19 goals.
