According to the Daily Mirror, the Swans are keen on the 25-year-old and are looking to spend £6.5m up front in a deal that will be highly incentivised.
It is believed the fee could reach as much as £9m depending on Swansea's presence in the Premier League over the next couple of seasons.
Gayle is down the Selhurst Park pecking order under Alan Pardew and will be given permission to leave if the South London club receives an appropriate offer.
The Swans are determined to land a new striker before Monday's transfer deadline and remain keen on Gayle while also stepping up their interest in Chievo forward Alberto Paloschi.
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!