The Iceland international will become Everton's club-record signing, surpassing the £31.8m the Toffees paid to bring Romelu Lukaku to the club
Sigurdsson was a key player for Paul Clement's side last season, scoring nine goals and laying on 13 assists to help the club beat the drop.
The 27-year-old is expected in Liverpool for a medical on Wednesday.
Everton have been chasing the player for over a year and will be delighted to finally get their man.
The Welsh side had originally been hoping to hold out for £50m for the former Tottenham midfielder but it appears that they were not able to do so.
Swansea had originally hoped to keep the player, who signed a new long-term contract last year making him the club's highest-paid player. However, a move looked almost inevitable after Sigurdsson was left out of the club's pre-season tour of the US "due to the current transfer speculation surrounding him".
It's a case of one in, one out for Everton after Gareth Barry completed his move to West Brom on Tuesday.