The two Premier League clubs have made progress in negotiations this week.
Although nothing has been signed yet, a deal is expected to be completed early next week.
Leicester have constantly made it clear they will not allow their players leave for low fees, and last summer, they made Manchester United pay £80m for Harry Maguire.
The former champions are not under pressure to sell, as Chilwell is under contract until 2024, but the situation could change if he wants to leave.
Chilwell is understood to be keen on the move with the opportunity for Champions League football at Chelsea.