The Toffees raided the Londoners 12 months ago to ring in highly-rated forward Ademola Lookman and scouts have alerted current boss Sam Allardyce to the qualities of 20-year-old Konsa.
He broke into Addicks boss Karl Robinson’s last season and has already made 29 appearances in all competitions so far this term.
Konsa, who was also a member of the England Under-20 team that won the World Cup last year, was linked with a move to Goodison Park in the summer when Ronald Koeman was at the helm but nothing materialised at the time.
However, reports in the south London press are claiming that a deal worth £4million will go through before the January window closes for business.
Rangers defender David Bates faces at least six weeks out with ankle ligament damage.