Dyche took over at Turf Moor in 2012 and has since lead Burnley to the Premier League and consolidated their place in England's top flight by avoiding the drop last season.
This campaign, Burnley have won at Stamford Bridge, earned draws away at Tottenham and Liverpool and, most recently, produced an unlikely victory at Goodison Park.
The results have enhanced Dyche's reputation to the point where Wright believes he is more than capable of taking the reigns at Arsenal.
However, the former Gunners forward knows that there is no way the club would entertain the idea.
'I believe he (Dyche) is somebody that needs to go, at some stage, to the next level,' Wright told the 'BBC'.
When asked if he could see Dyche in the hotseat at Arsenal he said: '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.'
Wright is a big fan of the former Chesterfield and Watford defender and believes if he doesn't end up in charge of a top club, he could end up succeeding Gareth Southgate as England boss.
'They'll probably start touting him for the England job,' Wright added.
'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.'