The Tigers, determined to keep hold of Clucas after the departure of seven key players already this summer, were reported to have turned down an initial bid for the 26-year-old last week.
A Hull spokesman confirmed to Press Association Sport that Burnley's latest approach had been rebuffed.
Clucas, a shining light for the Tigers last season as he played in all but one of their Premier League matches, had previously been linked with West Brom, who now appear set to sign Gareth Barry from Everton.
Tigers head coach Leonid Slutsky, appointed in June, is desperate to strengthen in his bid to guide the club straight back to the top flight.
The squad the Russian inherited has been depleted by the summer departures of defenders Curtis Davies, Harry Maguire, Josh Tymon and Andy Robertson, goalkeeper Eldin Jakupovic, plus midfielders Tom Huddlestone and Ahmed Elmohamady.
Should Clucas follow his former team-mates out of the KCOM Stadium, the departure of another prized asset would be a major blow to Slutsky's plans.
Source : PA