Details have not been officially announced but the contract is worth up to £42.5m with £20m guaranteed, according to ESPN.
"It feels good," Matthews told reporters. "It's funny, because not a whole lot has changed. I'm happy about it, but for me, it's nice, it's very humbling and it's a blessing, but it's business as usual for me.
"I'm glad that they have put their trust in me and belief in me in awarding me this, but at the same time you're going to get the same type of unwavering perseverance on the field and dedication that I'm going to bring day in and day out. I feel good but not much has changed."
Matthews has 204 combined tackles and 42.5 sacks in his career to date, which has been spent entirely at the Packers.