Koeman admitted desire to see a central defender, winger and striker arrive at Goodison Park.
Sigurdsson is eager to join Everton, who have two bids for the Icelandic international rejected by Swansea.
Ronald Koeman has told the club's board the Iceland international is key to reaching the Champions League.
Everton have been linked with a move for Sigurdsson throughout the summer.
In-demand Icelander will play no part in Saturday's match at St Andrew's, club confirm.
Toffees boss is hopeful of an agreement being reached with Swansea City amid reports deal is off.
Former Tottenham playmaker has rejoined first-team squad despite snubbing pre-season tour.
Tottenham Hotspur are also said to be interested in the 25-year-old midfielder.
Klaassen joined Everton from Ajax for around £24m earlier this summer.
Swans boss says all sides want to see protracted summer transfer saga resolved soon.
Owners Jason Levien and Steve Kaplan have released strong statement relating to interest in key playmaker.
Swans say that key playmaker "did not feel in the right frame of mind" to travel amid uncertainty over future.
Manager Paul Clement insists any sale will be dictated by Swansea amid rumours of £50m asking price.
Pat Nevin believes Champions League qualification could offset the potential £50m fee for the Icelander.
Antonio Conte "fuming" after losing out on Belgium international to Manchester United.
The 19-year-old striker rejected moves to Newcastle United and Brighton.
Terry's contract with Chelsea expires on 1 July.
Reports suggest Spurs will test Swansea's resolve with a £15m bid.
Sigurdsson has been on Everton's radar for a number of months.
The deal to sign Pickford broke the British transfer record for a goalkeeper.
Outcome of interest in influential duo vital as Paul Clement claims an overhaul is not required.
Foxes consigned to losing Mahrez as club line up Gylfi Sigurdsson as replacement.