Cillessen has 30 caps for the Netherlands
The Toffees, who will wave goodbye to Tim Howard this month, were linked with the 27-year-old earlier this year, when Roberto Martinez was still in charge at Goodison Park.
Howard's departure has left a vacancy in the Goodison goal
However, the Spaniard's departure has not affected Everton's interest, the report claims.
Securing a new goalkeeper is expected to be Koeman's first priority once his appointment is finally confirmed, with Joel Robles the only senior shot-stopper left on the Blues' books.
And Cillessen, who has two years left on his Ajax contract, is expected to cost in the region of £12 million.
Koeman is set to be named Everton's manager early next week
The former Nijmegen number 1 has 30 caps for Holland, and was tipped to become David de Gea's replacement when the Manchester United star was linked with Real Madrid.
England's Forster has also been linked
The Scottish tabloid also linked the 53-year-old with two more St. Mary's stars in defender Virgil van Dijk and midfielder Victor Wanyama.