Earlier this month, it was claimed that the Clarets had seen a bid of £5m rejected by the Tigers, who are keen to keep hold of the 26-year-old.
However, according to Sky Sports News, Sean Dyche has decided to double his initial offer to £10m as the Burnley boss looks to make use of the funds generated from the sales of Michael Keane and Andre Gray.
Hull signed Clucas for just £1.3m from Chesterfield in July 2015, but the Englishman was one of the few Tigers players to make a positive impact during their relegation campaign from the Premier League.
During his time at the KCOM Stadium, Clucas has netted nine times in 95 appearances in all competitions, while there are still two years remaining on the player's existing contract.
Despite their relegation, Hull are not in a position where they need to sell Clucas due to the money recouped from the sales of Eldin Jakupovic, Ahmed Elmohamady, Harry Maguire, Andrew Robertson and Tom Huddlestone.