As reported by David Ornstein of BBC Sport, Spurs have emerged as serious rivals to Chelsea for the veteran forward:
The Telegraph's John Percy reported on the medical:
According to Nizaar Kinsella of Goal, Tottenham are ready to pay £15 million for the Spain international and are hopeful of beating the Premier League champions to the player's signature.
Kinsella added that Tottenham's interest in the striker will contribute to what may be a "difficult" final day of the window for Chelsea.
However, Simon Johnson of the Evening Standard reported the Blues are the player's preferred destination should they agree a fee with Swansea.
As noted by journalist Kristof Terreur, the Blues seem to have narrowly missed out on a lot of their targets so far:
Llorente would be a solid addition for both sides, and it appears to be Spurs who have won the race. The former Juventus, Sevilla and Athletic Bilbao man was excellent for Swansea and a key factor in their Premier League survival last season.
While the 32-year-old is past his absolute best, he showed during 2016-17 he can still be a big asset in the final third. Llorente relishes physical battles and boasts the technique needed to hold the ball and bring others into play.
He's a beast in the air and also has a habit of netting in the biggest games, as noted by Squawka Football:
It's likely he will be a back-up option behind Harry Kane, the same role he'd enjoy behind Alvaro Morata at Stamford Bridge were he to join Chelsea.
At Chelsea, he would have had the familiarity of Antonio Conte. The Blues manager worked with the striker during their time together at Juventus and will be aware of the best way to get the most from the target man.
Still, Spurs are a tempting prospect. Mauricio Pochettino's side require a different option to Kane, and while Llorente may not be suited to the pressing style the manager seeks to utilise, he can provide an excellent focal point late in games when the team needs to go direct.