Mauricio Pochettino's side had the bid accepted by Swansea on Thursday morning [31 August] after agreeing to improve the terms previously offered by the Blues.
They have pipped London rivals Chelsea to the signing, as Llorente signed a two-year contract on transfer deadline day.
The Spanish striker has spent just the one season at SA1 since joining from Sevilla, scoring 15 goals to help the club beat Premier League relegation.
In 2015, he joined Ujpest from Zulte Waregem but a knee injury made him miss the majority of the 2016-17 season. Swans boss Paul Clement however, tends to favour a return to South Wales for Wilfried Bony who has failed to impress at Manchester City.
The 32-year-old left Athletic Bilbao, where he scored 118 goals in 333 appearances, for Turin in 2013, but after a promising first season with Juventus, his playing time rapidly diminished.
"That is considering that we have not replied to a single question in the past two weeks", was the comment from a Real source to MARCA after confirming the club had been bombarded with offers for the Spain Under 21 worldwide.