Sky Sports News understands there has been no official bid from Everton for the former Spain international at this stage, despite reports linking the Toffees with a move.
Llorente trained with Swansea on Wednesday but the Welsh club are lining up a move for Wilfried Bony, who left the club for Manchester City for £25m in 2015, if a deal does go ahead.
Swansea manager Paul Clement had said he was confident of keeping Llorente at the Liberty Stadium earlier this week but did admit "something can come out of absolutely nowhere" ahead of Deadline Day.
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