The full-back is currently on loan at Dutch giants PSV Eindhoven, who are sitting at the top of the Eredivisie following an unbeaten run of 20 matches.
Van Aanholt has featured in 16 top-flight contests - starting nine of those - with his only goal contribution coming as an assist in the 3-0 win over Go Ahead Eagles on September 27.
The Dutchman is most known in England for his spells at Sunderland and Crystal Palace, with the defender making over 200 appearances for the Black Cats and the Eagles combined.
According to Dutch outlet Eindhovens Dagblad, Burnley were hoping to bring Van Aanholt to Turf Moor in the final stages of the January transfer window.
The report claims that PSV were unwilling to allow the full-back to make a return to his parent club Galatasaray in order to seal a switch to Vincent Kompany's side.
It is understood that Van Aanholt's camp have grown frustrated with his lack of game time for PSV in recent months, despite the player beginning the campaign as first choice in his position.
The arrival of Sergino Dest on loan from La Liga champions Barcelona has led to the Dutchman falling down the pecking order on the left side of the defence.
Despite this, it is believed that PSV were not open to sending Van Aanholt back to Galatasaray, and Peter Bosz's men supposedly want to sign the 33-year-old on a permanent deal in the summer.
Burnley's hunt for a new left-back has been well documented since their promotion to the Premier League, with the Clarets missing out on Chelsea's Ian Maatsen on the final day of the summer transfer window.
The Clarets managed to recruit a right-back during the winter trading point, with 23-year-old Lorenz Assignon joining from Rennes on loan for the remainder of the season.
Assignon started Burnley's 2-2 draw with Fulham on the right side of the hosts' back four, meaning that natural right-back Vitinho moved into a left-back role.
Making his second substitute appearance since arriving on loan from Chelsea, David Datro Fofana bagged a brace to rescue a point for the relegation-threatened Clarets against Marco Silva's Cottagers.
The Ivorian's goals certainly lifted the mood at Turf Moor but Kompany's side are still seven points from safety with matches against Liverpool and Arsenal next on the agenda.