The Toffees are yet to replace Ronald Koeman, who was sacked two weeks ago, and Clarets boss Dyche is a frontrunner for the job.
But, with The Times reporting a two-part BBC Panorama programme about the club’s ownership is set to air, Barton believes he may be unwilling to put himself in a difficult situation and choose to sit tight instead.
“There is going to be a BBC Panorama programme on Sunday and Monday night which is about something which has gone on in the background of the club, in terms of who owns it. Usually if you get a Panorama interview done on you it’s not for positivity.
“I’m a mad keen Evertonian myself and this is not a great time for the football club. I don’t know what the programme is going to show but it doesn’t look great.
“If you’re Sean Dyche at Burnley, you know there are going to be job opportunities going forward. Burnley are a fantastically run football club and he’s a massive part of that. So is this the right time to go?
“Everton is a fantastic job to get but is it the job Sean Dyche would want to take at this moment in time? They’re in a relegation fight and there are issues on and off the pitch. Knowing the man as I do, he’s very much a risk averse guy and he likes to take his time over everything. He’s not one to jump at opportunities. He knows his quality and everybody is seeing that.
“I think he may well sit tight and ride out this Everton overture and he might wait for a better opportunity when he can get the chance to delve into the transfer market and put his stamp on a team in the summer. I think Burnley fans would wish him well if he left in the summer to another Premier League club.
“He knows his quality and he will be on a good wedge at Burnley, no doubt about it, although not the money he would get at one of the bigger clubs. At some point in time, though, it is going to happen. He’s a top class manager. For me, he’s the best I’ve worked with by a distance. It’s inevitable he’s going to go onto bigger and better things. If you do have quality then you are not looking to jump and move too quickly.
“He knows he has got a fantastic job at Burnley and they’re sitting seventh in the Premier League. How long can it be maintained? The future will show us that but why not wait to get a job at the start of a season where you can get your own culture and your own ethos stamped on a club. You’re going to take on someone else’s problems now.
“I think if Everton can get Sean Dyche through the door they’ve done incredibly well.
“He’s a modern day version of what David Moyes did to them. He would be a fantastic appointment for Everton.
“The difficulty lies, knowing the man as I do, in the off the pitch stuff could spook him and maybe make him sit tight and wait for a different opportunity.”