New Toffees boss is putting together his backroom team, but has ruled out another Southampton reunion.
Koeman, who was appointed as Toffees boss on a three-year deal on Tuesday, has so far brought brother Erwin and Jan Kluitenburg from previous club Southampton.
The Koeman brothers at previous club Southampton
The Dutch duo have assumed assistant manager and fitness coach duties respectively, though Everton remain on the hunt for a new goalkeeping tutor after Inaki Bergara followed Koeman's predecessor Roberto Martinez out of the club last month.
According to the Echo, the Toffees are aiming to make a swift appointment. However, the new man will not be Dave Watson, who is currently coaching England's goalkeepers at Euro 2016 and with whom the trio worked at St. Mary's.
When asked if Watson - not to be confused with the former Everton defender by the same name - would be next through the Goodison Park door, Koeman said: "No because that is not possible."
England coach Watson (L) is a former Barnsley goalkeeper
Former Toffees goalkeeper Neville Southall has also ruled himself out of the running on Twitter, posting: "Sorry not going to be next efc gk coach."
Sorry not going to be next efc gk coach— Neville Southall (@NevilleSouthall) 18 June 2016
Would like to mentor young gks though
Thanks for your thoughts
Earlier on Monday, the Toffees completed the signing of teenage Oldham Athletic goalkeeper for an undisclosed compensation fee.