The 27-year-old centre-back is David Moyes’ top target as the West Ham manager is keen to add defensive reinforcements to his squad ahead of next month’s transfer deadline.
Burnley do not want to lose the player, but the Clarets could accept an offer for the defender if it was impossible to turn down.
The Clarets knocked back an initial offer of £27m form West Ham and the Hammers also failed with another bid of £30m for Tarkowski.
West Ham made a third bid worth £31m, with the club guaranteeing a £25m fee up front but Burnley wasted little time in rejecting that offer as well.
The Hammers are now ready to end their interest in the Burnley defender and are prepared to consider other options.
It has been claimed that the east London club will not be making a fourth offer for the 27-year-old centre-back.
Tarkowski is open to a move to West Ham as he feels it could boost his chances of returning to the England squad.
But it seems the Hammers are now ready to pursue other targets after failing with three big-money bids for the Burnley centre-back.
