The former Atalanta defender arrived on loan in January with an option to buy and Milan have decided to invest €3.5m to acquire the centre-back.
The deadline of the option in his loan deal expires tomorrow, but Sky Sport Italia journalist claims the two sides have agreed on a permanent move for the Denmark international.
Kjaer has played 14 times in Serie A and has established himself as a vital member in Stefano Pioli’s side.
All the details have been sorted out and the report from journalist claims the 31-year-old will soon be announced as Milan’s latest signing.