In January, Ayew left West Ham United to secure a reunion with Swansea, a club he had previously represented during the 2015-16 campaign.
However, the 28-year-old failed to score in 12 appearances and with Swansea being relegated from the Premier League, the Welsh outfit are keen to get the attacker off the wage bill.
According to GhanaSoccerNet.com, Besiktas are looking to secure a loan agreement for Ayew and it appears that a temporary transfer is close to completion.
The report says that it remains unclear whether there is an option to purchase Ayew at the end of any spell in Turkey, although Besiktas will likely have to pay a large portion of the player's wages.
Ayew still has three years remaining on his contract at the Liberty Stadium.