Unai Emery’s men are fifth on the Premier League table with 16 points following their 2-2 draw against Crystal Palace on Sunday.
Arsenal scored two goals in the opening 10 minutes kind courtesy of their two center-halves, Sokratis Papastathopoulos and David Luiz, but Palace fought back to secure a draw.
The North Londoners have just two wins from their last five league games and questions have been raised as to whether the former Sevilla and Paris Saint Germain coach can guide Arsenal back into the top-four.
Winterburn, who made 440 league appearances for Arsenal between 1987 and 2000, believes Vieira should be considered as a potential successor to Emery.
“I would say keep an eye on what's happening with Patrick Vieira,” the 55-year-old told Gambling.com.
“If someone like that could prove they are able to run a club in the right way, then Arsenal may look towards that eventually.”
“I don't know if they would be as brave to go with someone like [Freddie] Ljungberg or [Mikel] Arteta.”
“They may be terrific coaches but when you take that manager's role it steps up another couple of levels again.”
Arsenal missed out on the top-four last season, and lost to Chelsea in the Europa League final at Baku but the former West Ham defender feels another season out of the top-four should cost Emery his job.
'I would think that if Arsenal don't get into the top four there will be a review,' he added.
'They can't get into the same position as with Arsene Wenger where 50 per cent of the supporters want him to stay and the other 50 per cent want him out and every time something goes wrong on the pitch it's almost as if the supporters are in crisis with each other.
'It wasn't pleasant sitting and watching those games at the very end of Arsene's reign. It was very uncomfortable as a former player to watch on from the sidelines.'
Arsenal face Liverpool at Anfield on Wednesday in the Carabao Cup before a Premier League match against Wolverhampton Wanderers at the weekend.