Charlton boss Karl Robinson, who played for both Everton and Liverpool in his youth, previously shared his belief that Konsa would be prised away in January, and said it would take a minimum of £5 million to land him.
"My gut feeling is that he will go in January," said the Liverpudlian, as quoted the Daily Mirror.
“If I was him I’d be looking to go in January but we should look for £5 million minimum. Maybe we could try for a loan back but the player has to want that."
Charlton have sold academy products Ademola Lookman and Joe Gomez to Everton and Liverpool, respectively, in recent years.
However, it is Everton who are said to be leading the race to sign Konsa, after Toffees scout Martyn Glover watched him in action against Blackburn last month.
The England Under-20 centre-back could be seen as the long-term successor to Ashley Williams and Phil Jagielka at Goodison Park, with both players well into their thirties.
Konsa has played 56 league games for Charlton, all of which have come in the third tier.
Are Everton the best club for Konsa?