The Toffees have pursued a deal to sign the 19 year old attacker in recent weeks, but a gap remains between Southampton's price tag and what they are willing to pay.
Saints have proposed a deal worth £45m plus £5m in add-ons and a 25 per cent sell-on clause after rejecting a number of bids from Everton.
The St Mary's side will not lower their valuation for the academy graduate who has a contract which runs until 2027.
The Athletic first reported that Everton are now "focusing on other targets, who are being pursued in parallel" after seeing their approach for Dibling hit a wall.
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Southampton, unsurprisingly, view Dibling as a player who can get them promoted back to the Premier League, and anticipate further interest this summer.
Saints negotiated the maximum price they could when selling winger Kamaldeen Sulemana to Atalanta earlier this summer.
That showcased their intent to sell players at the price they have set, which they also did ahead of the 2023/24 Championship season when they sold Romeo Lavia.
On selling Dibling this summer, manager Will Still explained on Saturday: "I have no idea (if he will depart).
"I think the club have set a price. We've got good players and there is interest for good players, which is normal."
Dibling played 60 minutes in an additional behind-closed-doors friendly at St Mary's against Premier League side Brighton on Saturday.