The move comes as Salah's camp suggested a decision on his next club could be close, with agent Ramy Abbas writing on social media: "We still don't know where Mohamed will play next season. But we may know very soon."
Salah left Liverpool this summer after nine years at Anfield, departing as a free agent following one of the most decorated individual careers in the club's history. In recent months he has been repeatedly linked with clubs in Saudi Arabia and MLS side Kansas City, though none of that interest has developed into anything concrete.
The emergence of Besiktas as serious suitors represents the clearest indication yet of where his next move could take him.
Salah was one of the standout performers at this summer's World Cup, inspiring Egypt to a memorable run that saw them push eventual finalists Argentina to the brink of an upset in the round of 16 before ultimately falling 3-2 having led by two goals.
His performances in the tournament only added to the intrigue surrounding his next destination, with elite clubs across multiple continents monitoring the situation closely.
For Besiktas, landing a player of Salah's stature would represent one of the most significant transfers in the club's history and a major statement of intent as they look to compete with Istanbul rivals Galatasaray and Fenerbahce, who have both been credited with interest in the Egyptian at various points this summer.
The Black Eagles have already been active in the market, recently strengthening their attacking options, and view a marquee free-agent capture as central to their ambitions of closing the gap on the division's traditional powers.
Talks between the two parties are continuing, though as things stand no final decision on his future appears to have been made, with Salah's representatives reportedly keen to ensure any move fits both his sporting and financial expectations before committing to a new chapter of his career.