The Toffees are targeting the Wales international as a replacement for seemingly Manchester City bound defender John Stones.
And they appeared to obtain his services earlier this week, with Wales Online reporting on Monday that a transfer to Goodison Park would go through within the next 48 hours.
However, the same publication later redacted, revealing the Liberty Stadium outfit had officially rejected a £10million bid from the Merseysiders.
But according to Sky Sports, Ronald Koeman’s side are now preparing to go back in with an improved offer for the 6 foot centre-half, who has made 346 appearances for his current employers.
And it’s further claimed that the Swans could be tempted to cash in on their 31-year-old captain, should Everton approach them with a bid around the £15million mark.
The Toffees are certainly in need of additions at the back after conceding a whopping 55 times in the Premier League last season.