The former Athletic Bilbao and Juventus forward has been targeted by the Swans as they seek to fill the void left by the summer departures of Eder, Alberto Paloschi and Bafetimbi Gomis.
The 31-year-old won three Serie A titles with Juventus and the Europa League with Sevilla last year, and has scored nearly 130 goals in his senior career.
But it is understood his game-time will be limited under new boss Jorge Sampaoli this season, and Sevilla's sporting director Monchi confirmed that negotiations for Llorente's transfer have made good progress.
"It has come a long way with the English club (Swansea) to reach an agreement, we expect to hear in the coming days," Monchi told the official Sevilla website.
Spanish media reports said on Tuesday that Llorente had agreed personal terms with Swansea and that the 24-times capped striker only had to tie up the loose ends of his Sevilla contract.
Swansea chairman Huw Jenkins also hinted his club were close to making "one or two additions" after what he admitted had been a frustrating summer on the transfer front.
"This pre-season has been difficult with the majority of European players returning late from the Euros and the timescale involved in the change of club ownership,'' Jenkins told the Swans' official website.
"I think it's been a frustrating summer so far for everyone, but I can assure you that every person connected to this football club has been working non-stop to get the squad right in all areas to start the season.
"It was always going to be the case after the Euros that a lot of the transfer business would be done later than normal.
"But we fully expect to make one or two additions in the next few days.
"We've worked hard in the summer to change things around for the forthcoming season.
"It was always going to be a process of getting players out first so that we were in a position to take advantage of the transfer window and bring new players in."
Swansea are also determined to keep skipper Ashley Williams having rejected a £10million offer from Everton for the 31-year-old Wales defender.
It remains to be seen whether Everton will return with an improved offer for the centre-back.
But Williams is set to return to pre-season training at Swansea this week following his extended break after Wales' success at Euro 2016.
Source : PA
Source: PA