Since joining Burnley in 2012, Dyche has experienced great success at Turf Moor, helping guide the club to two Premier League promotions - he is now the third longest serving manager in the division behind Arsene Wenger and Eddie Howe.
That came just months after Dyche was linked with the role at Palace, with Frank de Boer instead getting the position before being sacked just four league matches into the season.
"It's not something we're thinking about right now (shortlist of replacements), there is one person on our list and that is Sean Dyche". Burnley fans would be upset to see Dyche leave, a manager who arrived five years ago to the day when the team were mid-table in the Championship and steered them to two promotions, making Turf Moor a fortress in the process.
This clip is taken from 5 live Sport on Monday 30th October.
"Training is up there with the toughest I've had", he said. Everton are the next step.
"But because they're struggling this season, he hasn't been mentioned, because it's all about timing". He has guided Burnley to the Premier League in his second season in charge of the club.More news: Tiger Woods to return to competitive golf in Hero World Challenge
"He's doing well at Burnley, so it can only bode well for Everton if they get him but it could possibly derail Burnley if they lose a manager of that capacity".
Sean Dyche has Burnley well organised and hard to break down.
"When the iron is hot I think you've got to strike, and you want to get to the top". But respect has to be earned at all clubs and it has been earned here over a five-year period.
Everton moving for Dyche was an assumption made by many, although getting him and Bunrley to agree is quite another matter. He's obviously got ambition but we're happy he's here.
The Clarets have drastically improved on their away form from last season, already beating Chelsea at Stamford Bridge and earning draws against Tottenham Hotspur and Liverpool at Wembley and Anfield respectively.