Antonio Conte's side were reportedly close to agreeing a deal worth £15m, but Spurs have swooped in and offered a higher figure.
The Blues are expected to make a counter offer and The 32-year-old's final destination is understood to be too close to call.
Llorente scored 15 goals for Swansea last season and had previously played under Conte when the pair were at Juventus.
Swansea will only sanction the transfer if they can find a replacement before the window closes, with Manchester City striker Wilfried Bony a key target.