Benson impressed for the Belgian side last season, but he has only a year left on his contract at Antwerp, who he joined in 2019.
The winger is of interest to Burnley, with Vincent Kompany looking to rebuild the Clarets and ensure an instant return to the Premier League.
Burnley made an initial bid of £1.5m for Benson, which was turned down by the Belgian club.
They have now made a second proposal, according to Belgian journalist Sacha Tavolieri, but that too has been rejected as it is short of Antwerp’s valuation.
Discussions are still ongoing between the two clubs and it remains to be seen if Burnley go in with a third offer.
Benson established himself in the Antwerp team last season, making 34 appearances in the league and registering five goals along with six assists.
Burnley have strengthened their midfield and defence this summer but with some of their attackers being courted by Premier League clubs, they are looking to add depth the forward areas.
