The Derby County man had already been the subject of a £3.5million offer from the Clarets, who are looking to bolster their squad ahead of their new season in the Premier League.
But the 24-year-old isn’t going to be on the move just yet, with it believed Derby will also reject this bid, as they hold out for even more cash.
Hendrick was one of the star performers at EURO 2016 for the Republic of Ireland, helping them reach the knockout stages of the competition, only to lose 2-1 to hosts and eventual runners-up France. The man from Dublin featured in 33 Championship games for Derby last season, scoring twice.
Manager Sean Dyche has identified the midfielder as an area of his squad that needs addition, after losing Joey Barton to Rangers earlier on in the transfer window.