The Toffees are set for a big summer of spending under Koeman, as new investor Farhad Moshiri is willing to back the club in the transfer market as they seek new additions.
One area that must be strengthened by new boss Koeman is in goal. Long-serving Tim Howard has returned to the United States with Colorado Rapids, leaving the unconvincing Joel Robles as Everton's only senior goalkeeper ahead of the new Premier League season.
A number of names have been linked with moves to Goodison Park, but The Chronicle report that Everton are now stepping up their interest in signing Newcastle United ace Tim Krul.
Krul, 28, missed the majority of last season with a knee injury, but will be back for the start of the new season - when Newcastle will start life back in the Championship.
It's hard to see Dutch international Krul fancy playing in the second tier, and Everton are seemingly keen to exploit that by making a move to sign him as Howard's replacement this summer.
The report claims that new Everton manager Koeman is a big fan of Krul and will now step up his attempts to land the player, and the appointment of Dutch goalkeeping coach Patrick Lodewijks is noted as a possible draw for Krul.
Everton's interest is described as 'serious', with Koeman allegedly seeing Krul as the perfect replacement for Howard - and a bid is expected in the coming weeks.
Would you like to see Everton sign Krul this summer?