The Toffees are on the lookout for a new boss after parting ways with Ronald Koeman last week.
Everton have lost their two matches since the Dutchman’s departure and are now desperate to find a new manager with the club in the relegation zone.
And according to Sky Sports, Burnley boss Dyche is their first choice appointment.
They suggest the Merseyside outfit will make an approach for the 46-year-old in the next 24 hours.
Everton are set to make an approach for Sean Dyche in the next 24 hours
Everton Under 23 manager David Unsworth has taken the reins as interim manager for Everton’s defeats to Chelsea and Leicester and has expressed his interest in taking the role permanently.
But the Everton hierarchy look set to broaden their search for a new manager with Dyche firmly in their sights.
It remains to be seen whether Dyche will consider leaving Burnley having taken them to seventh in the league last night following their 1-0 win over Newcastle.
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Everton have lost both matches since sacking Ronald Koeman
"I just keep getting on with my job, simple as that," Dyche said when asked what he would do if a call came in from Everton.
"We just keep getting on with it. That's my focus and my team's focus. We have to do planning of course, but generally we take each game as it comes, we work on that, then we process the information, debrief it and we move forward again and keep that clarity of thought. That's how we keep going."
Dyche has been with Burnley since 2012, helping the club to two Premier League promotions.