Barry has been on the books of Everton on a permanent contract since 2014 following a loan spell at Goodison Park from Manchester City.
The English midfielder’s current contact with the Merseyside outfit runs until June 2018, according to Transfermarkt.co.uk.
According to The Daily Mail, Everton are ready to offer the former Aston Villa defensive midfielder a new one-year contract extension.
Barry made 23 starts in the Premier League last season, scoring two goals and providing one assist in the process.
It would be hard for the Englishman to play regularly this season due to competition for places in the Everton midfield.
Ronald Koeman’s side will be aiming to break into the top six of the Premier League this season and perhaps even claim a Champions League place.