The 44-year-old has emerged as one of the front runners to replace Ronald Koeman at Everton, after the Dutchman was sacked last week following a disastrous start to the campaign.
David Unsworth has been appointed as interim manager and Everton chairman Bill Kenwright is understood to be keen on him to land the job on a permanent basis, while the Toffees' major shareholder Farhad Moshiri is reportedly determined to bring in a more high-profile replacement for Koeman.
During his five years in charge of the Clarets, Dyche has emerged as an astute tactician and made some very smart signings and is generally accepted he has kept his side punching above their weight since they returned to the Premier League at the beginning of last season.
Dyche has admitted he has found it "nice" to be linked with the vacant managerial post at Goodison Park but insisted he remained focused on Burnley, who host Newcastle United on Monday night (30 October).
"I don't read much in the media, I know what I've said so I don't keep checking up on things," he told Sky Sports. "There have been some nice things said recently, some nice links, but the fact is I'm still here and still working hard."
However, according to the Sun, the Clarets' hierarchy has already lined up Moyes as replacement, should Dyche be lured to Merseyside.
Moyes, who started his managerial career at Burnley's local rivals Preston North End and still lives in the area, is reportedly seen by the Clarets board as the kind of steady hand they would require to replace Dyche, should he opt to move on.
The former Everton manager has been looking for a route back into football since he resigned at the end of last season, after his Sunderland side finished bottom of the table with just 24 points.
Relegation with Sunderland, the first of Moyes' managerial career, and being sacked by Real Sociedad after just 12 months in charge did little to boost the Scotsman's profile, which has never fully recovered since he endured a nightmare nine-month spell at Manchester United .
Moyes was a surprise choice to replace Sir Alex Ferguson at the helm, when the latter retired at the end of the 2012-13 season but, despite being given a six-year deal, he was sacked before the end of the season as United failed to qualify for a European competition for the first time since 1990.
However, during an 11-year stint at Goodison Park, Moyes established Everton as a top eight side despite a very modest budget.