The former Toffees defender talked up his old boss' chances, despite being asked about Mark Hughes.
Reports on Thursday link Mark Hughes with Roberto Martinez's post and, in an interview with Talksport, Watson said the move would be a good one for the Stoke boss.
However, the 42-year-old quickly moved on, suggesting his former manager Moyes is in with a shout having gained experience from difficult spells at Manchester United and Real Sociedad.
Asked if Hughes would consider Everton, Watson told the Alan Brazil Sports Breakfast show: "I’m sure he would. No disrespect to Stoke I would think that would be a very good move for Mark career-wise.
"This sounds crazy because there were people who were thinking Everton would need a change but I think Moyes did a fantastic job at Everton.
"He’ll have probably learned a lot from his time in Spain, his time at Man United - I maybe think that would be an option as well."
Moyes, who is said to be interesting Celtic and Aston Villa, moved to Goodison Park in 2002 from Preston.
After keeping the Toffees up, the Scot transformed them into regular top-half contenders in the Premier League despite having limited funds at his disposal.
Watson added: "Look at his progression from when he started he came into a team that was near the bottom, he immediately had an effect - made us well organised, stronger.
"My last season there we finished in a Champions League play-off place so he was improving. As I say with the experience he’s got now maybe he could go that little bit further, take them to the next level."
According to the Sun, Moyes prefers a return to Everton over any other club if Martinez is sacked.
However, the report received a lukewarm response from Toffees fans on Twitter, with many feeling his appointment would be a backwards step.
Martinez is under increasing pressure having failed to deliver a league win since March 1 and losing the FA Cup semi-final to Manchester United.