According to journalist Levent Ümit Erol, the Dutch defender's agent has already travelled to Turkey for formal talks with the club's hierarchy, with discussions centred on the terms of a potential move.
The 35-year-old is weighing up his options for the closing stages of his career after establishing himself as a modern-day great at Anfield, where he made 54 appearances last season, scoring eight goals and providing three assists.
Besiktas's leadership are said to be preparing an attractive proposal in an effort to convince their captain to embark on a fresh challenge in the Süper Lig.
Liverpool are understood to be following developments around their captain's camp closely as they plan for a summer of squad restructuring. While the club's hierarchy values his experience and continued influence in the dressing room, the financial terms on offer in Istanbul, along with the sporting project itself, are said to hold genuine appeal for those close to the player.
Van Dijk has been a cornerstone of Liverpool's defence since arriving from Southampton in January 2018, helping the club to a Premier League title and a Champions League triumph among other honours during his time on Merseyside. Any departure would represent a significant shift for a Liverpool side already navigating a period of transition, though no final decision on his future is expected imminently.
The Dutchman's standing as one of the finest centre-backs of his generation means any move away from Anfield would be treated as major news across European football, regardless of his age. Besiktas, for their part, see the addition of a player of Van Dijk's calibre and leadership qualities as a statement signing capable of transforming both their defensive solidity and their profile heading into the new campaign.
For now, talks remain at an early stage, with the coming weeks likely to prove decisive in determining whether Van Dijk opts to see out the remainder of his career in the Premier League or embarks on a new challenge in Turkey.