Clucas moved to the KCOM Stadium from Chesterfield in the summer of 2015 and signed a three-year deal with the Tigers.
The 26-year-old was part of the side that gained promotion to the Premier League in 2015-16, scoring six goals from 44 Championship appearances for the Yorkshire outfit.
He enjoyed 37 Premier League outings last term and scored three goals but could not prevent Hull slipping back into the second tier of English football.
The Lincoln-born schemer played for Hull in their opening 1-1 draw against Aston Villa but, according to reports, is now on the Clarets' radar as Sean Dyche looks to reinvest some of the money from sale of Andre Gray to Watford.
However, it has been suggested that Tigers chief Leonid Slutsky is unwilling to part with the player as he wants to keep his best players at the club despite the fear that Clucas could be tempted by a quick return to the top flight.