Spurs have moved late to gazump London rivals Chelsea in the race for Llorente.
While a move to Tottenham is not cut and dry, the Blues face missing out unless they act quickly.
According to the Daily Mail, Swansea have accepted an offer from Mauricio Pochettino's side.
Llorente is said to be weighing up his future with his brother, who acts as his agent.
The move is still up in the air at this stage and 32-year-old Llorente will make the final decision.
But his form in his debut season in the Premier League has clearly attracted some of England's elite clubs.
Llorente's relationship remains strong with Chelsea boss Antonio Conte, but a switch to Spurs may appeal too.
Conte will be furious if Chelsea do indeed miss out on another key target.
The Italian saw Manchester United snatch Romelu Lukaku from under their noses earlier in the window.
And ahead of tonight's 11pm deadline, it appears Spurs had a late bid accepted for striker Llorente and offered him a more lucrative deal.
