Phil Kirkbride of the Liverpool Echo reported the Toffees have taken a crucial step toward landing the attacking midfielder, who is described as "keen on a move to England."
Klaassen scored 20 goals and made 12 assists in 50 appearances for the Dutch giants across all competitions last season, where Ajax finished as runners-up to Manchester United in the UEFA Europa League final after a 2-0 defeat.
Klaassen could become Everton's second major signing of the summer after Sunderland goalkeeper Jordan Pickford, who will undergo a medical on Thursday ahead of a £30 million move, per the Liverpool Echo's David Prentice.
Merseyside reporter Richard Buxton hailed the out-of-character initiative already being shown by Everton this summer, with the club usually left waiting to conduct significant business in the transfer window:
The 24-year-old missed only one of Ajax's league fixtures in 2016-17, which came due to suspension, and the fact he started in all but five of his 50 outings proves what an important member of the setup he was in Amsterdam.
While Ajax may be saddened to lose one of their most promising servants, Dutch football writer Michiel Jongsma highlighted the good value they're getting out of the reported deal:
Toffees manager Ronald Koeman has already allowed Tom Cleverley to join Watford on a permanent deal this summer, and Klaassen looks as though he could be a stellar midfield replacement.
Despite his age, Klaassen has already earned 14 caps for the Netherlands and scored four international goals in the process, establishing himself as a consistent starter for De Oranje over the past year or so.
The arrival of another attacking midfield force could spell the end of Ross Barkley's Everton stay, however. The Mirror's David Maddock also reported Swansea City's Gylfi Sigurdsson is a target, pushing Barkley out of Koeman's plans.
Klaassen will look to form a dynamic Dutch duo under compatriot Koeman's tutelage as a deal for the Ajax talisman closes within reach, with only a transfer fee left to be agreed between the two clubs.