The Blues were leading the chase for the Spanish forward and had even held talks earlier in the week, but Mauricio Pochettino's side have made a late bid to trump their London rivals.
According to Sky Sports, Tottenham are now in pole position to sign the Llorente but Chelsea are unwilling to give up on their chase for the forward and will continue to push to bring him to Stamford Bridge.
The south Wales club are said to be demanding a fee of around €10m - €15m (£13.8m) from interested clubs and despite Chelsea being in talks for weeks, Spurs have had their bid accepted ahead of the Blues. The 32-year-old striker is seen as a backup to Harry Kane, while Chelsea want him as an alternative to Alvaro Morata.
Llorente joined Swansea last summer and proved to be a shrewd buy after he managed 15 goals in the Premier League to help the club steer clear of the relegation battle in the second half of the season. The only advantage Chelsea have in luring the Spaniard to Stamford Bridge is his relationship with Antonio Conte, under whom he played during his stint at Juventus.
Pochettino revealed that Tottenham will make at least three signings before deadline day following the arrival of Davinson Sanchez from Ajax and looks like they are on course after they announced the signing of Juan Foyth on Wednesday (30 August). They are close to signing Serge Aurier from Paris Saint-Germain and Fernando Llorente will make it three additions in two days.
Meanwhile, Kane's current backup Vincent Janssen has refused to rule out a potential exit from Tottenham before Thursday's transfer deadline. The Dutch striker, who is currently on international duty has admitted that being an option from the bench is not ideal as he wants to play regular football.
The 23-year-old made 27 appearances in the league last season of which just seven were from the start and he managed just two goals, which is unlikely to get him more game time. He has been linked with a move to West Bromwich Albion and Llorente's arrival could hasten his departure.