Dyche is seen as one of the frontrunners to succeed Ronald Koeman, who was relieved of his duties on Monday (23 October) after overseeing a dreadful start to the season on Merseyside. Under-23 boss David Unsworth has stepped into the breach for the time being, and the former Toffees full-back watched over a much-improved performance as his side lost 2-1 to Chelsea in the fourth round of the Carabao Cup on Wednesday.
Unsworth has made no secret of his desire to assume the role full-time, and a number of Everton's squad want the bosses at Goodison to make that a reality, according to the Independent. 'Rhino' will be in charge of Everton when they travel to face Leicester City, who recently appointed Claude Puel as Craig Shakespeare's successor, on Sunday, while Dyche will be getting his Burnley side ready for their clash against newly-promoted Newcastle United.
Many people would have had their head turned by speculation surrounding their future, but Dyche, who was heavily linked with the Crystal Palace job in the summer, insists he has not been taken aback by the rumours linking him with Everton.
"I've just continued to get on with what I do here, and that's the focus," Dyche said in his press conference, relayed by Sky Sports. "I've had the experience of those sorts of links at various times - more so in the last couple of years.
"You're always flattered because it recognises the work that myself, my players, my staff do."
It is not known how long Dyche has left to run on his current contract with Burnley, but the former Millwall centre-back admits he did not expect to be in charge at Turf Moor for five years. His reign may well come to an end soon if Everton decide to make a move, but Dyche will be wary of the demands placed on him if he does decide to succeed Koeman.
"In the modern world of football, I'm surprised I've been here five years. There are so many differentials now that you never know," Dyche said.
"The demands now are far higher. The biggest thing is the misalignment of expectation. There's such a divide at clubs between reality and what they think is reality.
"When I first got here, I worked really hard through the media to tell the truth against the perception of what the club was."