The aforementioned source claims that a deal had been verbally agreed before Everton withdrew the offer and referred the Swans back to their previous £40m bid.
Everton officials fear that this may complicate the deal further and prevent the deal from going through as expected.
Both Everton boss Ronaldo Koeman and Swansea manager Paul Clement have planned their future business on the assumption that this deal would go through, meaning that the two clubs are likely to relent and push through a deal.
The two clubs continue to negotiate, with 'Sky Sports News' under the impression that there is less than £5m between the two sides' valuations.
Following Everton's friendly against Sevilla on Sunday Koeman said that the 'Toffees' were close to a deal.
"We are close [on Sigurdsson], we have to wait. But still, the deal is not done," he told the press.
Swansea boss Paul Clement on the other hand said that there had been no new developments, saying: "No change since we last spoke. All parties would like it resolved."
Sigurdsson starred for the Swansea last season, scoring 10 goals in 40 games in all competitions as they narrowly escaped relegation.