Clarets boss Vincent Kompany, who got an up close look at Flemming in the Championship last season, has seen enough to badly want him at Turf Moor.
Burnley have been trying to deliver for their manager, but so far no agreement with Millwall is in place.
It has emerged that Burnley have now failed with four offers for Flemming, with none of the four being good enough for Millwall to accept.
And there is no release clause in Flemming’s contract that the Clarets could look to trigger.
The 24-year-old attacker cost Millwall a fee of £1.7m to land from Dutch outfit Fortuna Sittard and is expected to cost Burnley in excess of £8m if he does move north.
Flemming came through the youth set-up at Ajax, but had to move to Zwolle to make the breakthrough into the Dutch top flight.
He joined Millwall just last summer, penning a three-year deal, and found the back of the net 15 times for Gary Rowett’s men last term.
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