Burnley snapped Weghorst from German side Wolfsburg in the January transfer window in a bid to boost their chances of avoiding relegation from the Premier League.
The Clarets could not beat the drop however and Weghorst does not want to play his football in the Championship next term.
He is now poised to complete a season-long loan move to Besiktas and the Turkish side are expected to hold an option to buy in the deal.
Besiktas are waiting for paperwork to be sent over by Burnley, according to Turkish journalist Ertan Suzgun, as they look to complete the deal.
The Turkish side are hoping to be able to bring Weghorst to Istanbul on Monday.
Besiktas finished just sixth in the Turkish Super Lig last term and are looking to push up to challenge for the title next term, something they hope Weghorst will be able to help with.
Weghorst has been prolific in both the Eredivisie and the Bundesliga, grabbing 51 goals in the Dutch top flight and 59 in the German top flight.