Everton have already signed a playmaker in Davy Klaassen this summer but Koeman is looking to add more depth to that position.
According to Daily Mirror, the Merseyside outfit are looking to sign the Swansea midfielder Gylfi Sigurdsson and have already submitted a £30 million bid for the Iceland international.
The report adds that Swansea have rejected the offer and it seems that the Welsh club will only sell the player if their valuation is met. It will be interesting to see if Koeman returns with an improved offer now.
The Toffees have already splashed out on the likes of Pickford and Klaassen. They will need to improve their back four if they want to challenge for the European places and therefore spending in excess of £30 million for another midfielder might not be a wise choice.
Having said that, Sigurdsson is a very good player and he would be a major upgrade on Ross Barkley right now. The England midfielder has been linked with a move away from Goodison Park this summer.
In other news, Daily Mirror are also reporting that Everton are set to announce the signing of Sandro Ramirez soon.
The Malaga forward will join up with his new teammates once the European Under-21 Championship is over.
The Premier League outfit have agreed a £5.2 million deal for Ramirez.