John had been linked with a loan move from Rangers, but could now depart for good just eight months after he signed a three-year deal at Ibrox.
The Wales international, who began his career at Swansea's rivals Cardiff City, looks to have fallen down the pecking order at Rangers, who have signed Jon Flanagan and Borna Barisic in his position this summer.
And it is claimed he could be allowed to leave in search of regular football.
John may not be the only Rangers player leaving Ibrox on Thursday, amid strong interest in James Tavernier (West Bromwich Albion and Brighton and Hove Albion) and Josh Windass (Ipswich Town and Wigan Athletic).
Whether the departures of Tavernier and Windass are sanctioned remains to be seen, however, with both players featuring more often than John has under Steven Gerrard this season.
Would Rangers be making a mistake selling John to Swansea?