It was believed a move was within touching distance earlier in the week when Toffees boss Ronald Koeman said a deal was "close" following his side's 2-2 friendly draw against Sevilla on Sunday.
Swansea have been holding out for up to £50million, while Everton have had several offers of around £40m rebuffed by their Premier League rivals.
It seemed the Merseysiders had edged their bid nearer to £45m and that looked to have done the trick in persuading the Swans to sell.
But reports late on Tuesday claimed Everton have withdrawn the offer and referred back to a previous bid, understood to be around £40m.
The two sides are again around £5million apart in their valuation, while personal terms should not be an issue should they finally agree on a price given that Sigurdsson has made no secret of his desire to move to Everton.
A deal is not thought to be dead in the water just yet, but Swans boss Paul Clement has made it known that he wants the situation resolved as soon as possible to give him plenty of time to find a suitable replacement for the Iceland international before the August 31 transfer deadline.