The Frenchman has been in charge of first-team affairs at the North London club since 1996, delivering three Premier League titles and seven FA Cups.
Last season, however, was the first in which he failed to guide to Champions League qualification and calls for him to be sacked have intensified.
Numerous protests have taken place at the Emirates in a bid for him to be ousted as Arsenal boss, though despite last season's failings and pressure from the Gunners support, Wenger was handed a fresh two-year deal at the end of last term.
The newest deal is likely to be Wenger's last at the club, with a number of names already being talked about to take charge of Arsenal once the 67-year-old retires from management.
Bournemouth's Eddie Howe was made favourite last season, though his difficult start to the 2016-17 Premier League season has seen him distanced from the job.
Former Borussia Dortmund head coach Thomas Tuchel was also amongst the front runners for the hot-seat, with Carlo Ancelotti reportedly expressing his interest following his dismissal from Bayern Munich last month.
Former Arsenal favourite Wright, however, has marked Burnley boss Dyche as Wenger's replacement following his success at Turf Moor.
"I believe he is somebody that needs to go, at some stage, to the next level," Wright told BBC Radio 5 live.
Asked if Wright would opt for Dyche as Wenger's replacement, the former striker 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."
Dyche led Burnley to promotion to the Premier League in 2013-14, and repeated the feat as Championship title winners in 2015-16, before keeping the club in the English top-flight ahead of the current season.