The Ivory Coast international is not in Man United's plans moving forward and will therefore leave the club on a permanent basis before the summer transfer window closes for business.
There is believed to have been a host of Saudi Arabian interest in the centre-back, and a recent report claimed that the 29-year-old had received two offers from the Saudi Pro League.
However, according to transfer expert Fabrizio Romano, Besiktas are now in pole position, having reached an agreement with the centre-back over personal terms.
Romano claims that the two clubs will now hold discussions in the hope of coming to an agreement over a fee.
Bailly spent the 2022-23 campaign on loan at French side Marseille, making 23 appearances for the club before returning to his parent club over the summer.
The Ivorian has not been involved in Man United's preparations for the new campaign, though, with Ten Hag making it clear that he is not a part of his thoughts for the 2023-24 season.
Bailly arrived at Old Trafford from Villarreal in the summer of 2016, and he made 38 appearances for the Red Devils during an impressive 2016-17 campaign.
The centre-back has struggled with a host of injury problems since, though, and he has only made 113 appearances for the club in total over the space of six seasons.
Bailly has won silverware during his time at Old Trafford, lifting the EFL Cup, Community Shield and Europa League, while he won the Africa Cup of Nations with Ivory Coast in 2015.
Man United are also believed to be on the verge of selling Brazilian midfielder Fred to Fenerbahce.
Meanwhile, it has been claimed that the Istanbul giants have been offered the chance to sign Red Devils forward Anthony Martial on a free transfer this summer.
Axel Tuanzebe and Phil Jones left Old Trafford on free transfers earlier this summer, while there is expected to be another centre-back departure in the near future, with Harry Maguire seemingly closing in on a switch to West Ham United.
However, Ten Hag said during his press conference on Friday that Maguire would be available for Monday's Premier League opener against Wolverhampton Wanderers.