Sigurdsson has been high on Everton manager Ronald Koeman’s wanted list for several months, but protracted negotiations over the transfer fee had delayed the Iceland international’s move.
The former Tottenham player missed Swansea’s pre-season tour of the United States while the talks dragged on.
He rejoined first-team training after Swansea’s return from the States, but he did not figure in their Premier League opener at Southampton on Saturday.
Everton were keen to tie up the swoop before the transfer window closes at the end of August and it is believed Swansea have at last agreed to sell their prize asset.
The 27-year-old is set to undergo a medical on Wednesday prior to joining Everton for what will be the Merseyside club’s record transfer fee.
Everton’s previous highest fee was £31 million for Romelu Lukaku, who was sold to Manchester United for a British record £75 million during the close-season.
Koeman has used the Lukaku windfall to fund a spending spree that will surpass £100 million once Sigurdsson signs.
The Dutchman re-signed Wayne Rooney from Manchester United, while also bringing in Michael Keane, Jordan Pickford, Davy Klaassen and Sandro Ramirez.
Sigurdsson scored nine goals and contributed 13 assists last season as Swansea climbed out of the relegation zone to ensure their survival.
His arrival may herald the imminent departure of England midfielder Ross Barkley, who has refused to sign a new contract at Everton and is keen to leave, with Tottenham among the clubs interested in his signature.
Swansea boss Paul Clement will use the Sigurdsson cash to strengthen his squad with two or three new signings.
Manchester City striker Wilfried Bony, West Brom forward Nacer Chadli and Stoke midfielder Joe Allen have all been linked with Swansea.
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