The Clarets made a solid start to life back in the Championship on Friday night when they beat Huddersfield Town in the opening game of the season.
Vincent Kompany was pleased with the performance, but is pushing for more additions in the coming weeks of the window.
Benson is a big target for him and Anderlecht have already rejected two bids from the Clarets for the Belgian winger.
And according to Belgian journalist Sacha Tavolieri, Burnley put in a fresh bid earlier this morning to further push to sign the winger.
It has been claimed Anderlecht have received an offer of €4m, including add-ons, from Burnley for Benson.
The Championship club are hopeful that Anderlecht will accept the offer and reach an agreement for the winger’s sale.
Benson recently addressed the interest from Burnley and is believed to be open to a move to Turf Moor to play under the former Anderlecht boss.
