The Wales international moved to Parkhead from Cardiff in 2011 and was highlighted in midweek by Neil Lennon as a player that the SPL champions were keen to hold on to.
|11/1||Celtic are 11/1 with Bet365 to beat Motherwell 3-0 at Fir Park on Saturday
Matthews told Sky Sports News: "The club has been great to me and I really enjoy being here so if I am offered a new deal I will happily sign, definitely."
Matthews claimed the promise of playing regular Champions League football was behind his desire to secure his future at Celtic Park.
"The club is massive," he added. "The Champions League is one of the main things, obviously, but the fans are brilliant and it is an honour to play in front of them every week so I am happy to stay.
"Every player wants to play in the Champions League but I think even without it Celtic are a massive club, so I am happy here.
"The atmosphere against Benfica [last week] was probably one of the best that I have played in. It was great to be a part of that so I am looking forward to the rest of the campaign."
The Bhoys will look to build on their midweek 4-1 win over Raith Rovers in the Scottish League Cup when they face SPL table-toppers Motherwell on Saturday.