The former Manchester United man was sacked by Valencia on Wednesday.
The La Liga side sacked the former Manchester United man on Wednesday following a less than impressive time in charge at the Mestalla.
He managed just 10 wins during his time in Spain, leaving the side just six points of the relegation zone.
It is now unknown whether he will return to his former job as a pundit for Sky Sports, or seek another managers job elsewhere.
If it is the former, Savage says he will have to lay off Arsene Wenger at Arsenal having experienced the manager’s job first hand.
“Knowing Gary as I do, he will accept that Valencia’s results were not good enough,” he told the Mirror.
“But this will only make him even more fiercely determined to be a success at his next job. He’s gone to the top in everything else he’s ever tried, so don’t rule him out.”
“But if he goes back to the gadgets and tools of a pundit’s trade, he may have to lay off calling Arsene Wenger’s tactics “naive” for a while after tasting life on the other side of the fence!”
For now, Neville will surely be focused on the national side as he gears up to travel to Euro 2016 as part of Roy Hodgson’s coaching set up.
His stint at Valencia doesn’t appear to have hurt his reputation, with many believing the job was always going to be a hard one for a first time manager such as him.
After such a high profile job though, perhaps starting further down the ladder would be a wiser move should he wish to continue as a manager in future.