Having joined the Lancashire club in 2012, Dyche has led Burnley to two Premier League promotions and one relegation in the last five seasons.
"I believe he (Dyche) is somebody that needs to go, at some stage, to the next level," Wright told BBC Radio 5 live.
"He's an experienced manager in the Premier League. He's done his groundwork with relatively smaller teams and got them set up well."
Arsene Wenger signed a new two-year deal at Arsenal after intense media speculation surrounding the Frenchman's future towards the end of last season.
Asked if Dyche is a suitable replacement for Wenger at Arsenal, Wright replied: "Yeah. The fact is would they give Sean Dyche that job?
"In respect to how his team sets up when they're defending, he's obviously got acumen - but will he get a job like that? I don't think he will.
"They'll probably start touting him for the England job. We're so bereft of any manager that's good enough, that can take on the England job. If he carries on like he is he'll probably be pushed into that role."
(Reporting by Hardik Vyas in Bengaluru; editing by Sudipto Ganguly)
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