The Championship side had an initial offer worth up to £1.5 million rejected for the winger.
But Royal Antwerp are hoping to hold out for almost double that figure, with £2.7 million being considered the player’s price tag.
That is the fee that the Belgian side paid to sign the 25-year old in 2019 from Genk.
Benson has only 12 months remaining on his current contract with Antwerp, but the club is still hopeful it can recoup the fee it paid for the player three years ago.
Burnley may be in the market for a new winger, depending on the future of Maxwel Cornet.
The former Lyon player has a release clause reportedly worth up to £17.5 million and is expected to depart this summer given the intense transfer speculation surrounding the winger.
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Burnley have already made several additions to the side this summer following the appointment of Vincent Kompany as the latest permanent manager.
The Belgian will undertake his first competitive match in charge of the team on Friday night when they face Huddersfield Town.
If Benson is their chosen target to replace Cornet, then the fee of £2.7 million should not be seen as too big of an obstacle in order to get the deal over the line.
It would be a much lower fee than the one they will likely receive for the sale of Cornet, so there is still a solid profit there for this summer.
Benson has also attracted interest from the likes of Inter Milan and AS Monaco in the past, showing he is clearly a very talented player.
Kompany will also have experienced his talent first hand from his time at Anderlecht so should have a good idea of what kind of player he is targeting.