The 26-year-old defender has one year left on his current deal at the Swans who still have an outside chance of playing in the Europa League next season.
Taylor, a Wales international, has been at the Liberty Stadium since July 2010, having moved to south Wales from Wrexham.
Sky sources also understand that they will be seeking another left back during August’s transfer window.
Watch Swansea take on Arsenal live on Sky Sports 1 HD on Monday. Kick-off at The Emirates, 8pm.