The 28-year-old international currently players for Kasimpasa in Turkey, having moved on a free transfer from Ajax in 2013. He made 29 appearances for the club this term, scoring nine goals and registering five assists in the Super Lig.
He has also earned 42 caps for the Netherlands, but he is most well known in England for his four year stay with Liverpool, whom he signed for in 2007, before leaving in 2011 after years of struggle.
Arriving from Ajax with a great reputation, he struggled to make it at Anfield, scoring 22 goals and registering 18 assists in 146 games.
These struggles saw him leave the club for Hoffenheim, but Swansea are now keen on bringing him back to England this summer.
The report claims that, although Besiktas look favourites for his signature, Garry Monk is preparing to move for Babel, as he looks to strengthen the wing positions ahead of next season.
The Welsh club have shown an interest in Babel in the past, attempting to sign him in January 2013, months prior to his exit from Ajax. However, no deal could be struck, and they want to return for his services this summer.