The former Manchester United midfielder has been out of action since sustaining an injury in October but could be back for the last game of the season
Gibson sustained a knee ligament injury while on international duty with Ireland against Kazakhstan in October 2013 and has been on the sidelines since, but Everton boss Roberto Martinez has set a target for the midfielder's return.
Speaking to reporters ahead of the Toffees' Premier League clash against West Ham, Martinez explained that Gibson and the club's other long-term injury concern, Arouna Kone, could be back in contention by May.
"Darron has always been ahead of the expected progress. So he has a small chance," said Martinez. "Even if it's just making the final home game it would be terrific. He is running now and progressing well. Darron is the more likely of the two.
"Arouna is also progressing well and if we could see him on that last day it would be great. From a medical point of view we shouldn't really expect either to be back before the end of the season.
"But to maybe have them back for the Manchester City game [on May 3] would end what has been a disappointing season for them both personally on a high note. They're both important to our set-up."