The Turks are in ongoing contacts with Feyenoord for Danish international and are trying to push through a deal.
Jorgensen, 29, was the Eredivisie top scorer in his first season with Feyenoord with 21 goals.
In the three seasons since, goals have not been quite as common, but he remains a prolific marksman.
Besiktas want to close the gap on Süper Lig champions Basaksehir next season after finishing seven points behind them.
Jorgensen has scored a total of 53 goals in 122 goals since joining Feyenoord from FC Copenhagen in 2016.