The 27-year-old, due to sign a five year contract worth £100,000 a week, would be Everton’s record signing, topping the £31.8m they paid Chelsea for Romelu Lukaku in 2014.
Sigurdsson, who played a pivotal role in Swansea retaining their Premier League status last term after scoring nine goals and creating 13, pulled out of their pre-season tour to the United States and was not included in the squad for last weekend’s 0-0 draw with Southampton.
Swansea, who are also battling to keep Chelsea target and veteran Spanish striker Fernando Llorente, had wanted £50m.
They have already turned down two bids from Everton and rejected a reported £40m offer one from 2016 champions Leicester at the beginning of the summer.
It is believed, though, that Everton will get their way in paying an initial £40m with another £5m to follow, taking their close season spending under Ronald Koeman to over £140m.
Everton’s majority shareholder Farhad Moshiri has spent close to £100m so far in recruiting former favourite Wayne Rooney, goalkeeper Jordan Pickford (£30m), midfielder Davy Klaassen (£23.6m), defender Michael Keane (£30m), Cuco Martina and Sandro Ramirez.
However, a considerable proportion of the spending has been financed by the £75m sale of striker Romelu Lukaku to Manchester United.
They could further balance the books with a big money sale of midfielder Ross Barkley to Tottenham Hotspur, although he is presently injured and likely to be out till October.
Icelandic international Sigurdsson finished third in combined goals and assists amongst midfielders in the Premier League last season behind Manchester City’s Kevin de Bruyne and Tottenham’s Christian Eriksen.
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