Both Leicester and Everton are understood to have seen £40m bids turned down by the Swans already this month, with Paul Clement's side refusing to lower their £50m asking price for the Iceland international.
The Leicester Mercury reports that the Foxes had been considering returning with a second bid which would match that valuation, but that the player himself is eager to join Everton instead.
The 27-year-old played in Swansea's pre-season friendly defeat to Barnet earlier this week but then pulled out of the tour to the USA due to "uncertainty over his future".
Sigurdsson is understood to have been convinced by Everton's activity already during this summer's transfer window, with the Toffees having signed six new players, including Wayne Rooney, Jordan Pickford, Michael Keane and Davy Klaassen.
Sigurdsson scored nine goals and created 13 more to help Swansea retain their Premier League status last season.