The 20-year-old signed a four-year deal at Parkhead just 12 months ago, but he now wants to secure his future even further.
Matthews immediately became a regular for Neil Lennon after he moved to Parkhead from Cardiff City last summer. His impressive form even saw him linked with Premier League clubs such as Everton and Fulham, but the Welshman has said that he wants to stay in Scotland.
When asked about his future, the right-back said: "A new contract? It's massive for me. What will happen will happen but I'm happy at the club.
"It has definitely gone better than I could have expected. When I signed I hadn't been playing regularly for Cardiff and yet I've managed to establish myself here as first-choice right-back.
"If I get the offer I'll be happy to stay because this is a great club. Hopefully I showed how well I could do last season and I've carried that form into this season."
Matthews has earned nine caps for Wales since making his debut in 2011.