Dyche was sacked by the Clarets ahead of their final eight games of the season with the club 18th in the Premier League and four points from safety.
Next up for the for Burnley is West Ham, who were recently managed by former player Bilic, who is eyeing a return to the dugout.
And that could be with the Turf Moor club who have a managerial vacancy for the first time in more than nine years.
The 53-year-old saw his stock rise massively during a six-year stint with the Croatian national team between 2006 and 2012, later joining West Ham in 2015.
A star defender with the club in the 1990s, Bilic guided the club to a seventh placed league finish, setting a number of Premier League records for West Ham, before his departure in 2017.
Bilic later took up the reins at West Brom, guiding the club to top flight promotion in 2020, and he last managed Beijing Guoan as recently as this January.
Bilic is among those linked with the Burnley job, along with the likes of Sam Allardyce, Wayne Rooney and Chris Wilder.
Allardyce even replaced Bilic at West Brom at the tail end of 2020, but left the club at the end of the 2020/21 campaign as they were relegated to the Championship.
Burnley wouldn't be a shock move for the ex-England manager given spells with Preston and Bolton as a player and the latter as a coach, along with Blackburn, with all those sides residing in Lancashire.