Speculation has been rife about the future of the 26-year-old Hoops midfielder, whose current deal expires at the end of the season.
Crystal Palace are reportedly set to make a £600,000 bid for the former Cardiff player when the transfer window opens on Wednesday but Matthews is positive the player and the club will rebuff the south Londoners' interest.
"I think he will stay," Matthews said. "I am hoping so. He is a great player for us and a good mate of mine so I am sure if they give him the right contract he will sign. He loves it here, his family loves it up here.
"I can't really ask him about it, not really. It is up to him and his family, whatever he thinks is right for him.
"But I hope he does stay and I'm sure Celtic fans hope he stays because he has been a great player for us."
Matthews, 21, is back in the side after recovering from a fever which came after a broken collarbone had kept him on the sidelines for over two months.
The former Bluebirds full-back played the full 90 minutes in the 1-0 win over Inverness Caledonian Thistle at the Tulloch Caledonian Stadium on Sunday and is looking to help the Hoops extend their unbeaten run in the league to 19 games.
"Yes, it is a big thing," he said. "We want to keep it going for as long as possible. We know it is going to be tough because we know everyone is going to try to break the record but we just have to play the way we are playing and take our chances when they come and hopefully keep it going.
"Every derby is tough and Partick play good football so we know we have got to be on top of our game to keep our run going.
"But it will be good to be back at Celtic Park, the fans will be brilliant like they always are. It will be nice to play on a nice pitch and get our passing going which we couldn't do against Inverness.
"I don't feel fully fit but I am getting there, my match-sharpness is getting better so hopefully I can get a run in the team, keep my fitness going and get back to where I was before my injury."