The experienced centre-back is out of contract at Napoli in the summer and could put pen to paper to a pre-contractual agreement with a foreign club this month.
Napoli could also be tested with offers for Maksimovic, who has been linked with reuniting with the club’s former boss Carlo Ancelotti at Everton, in the current window.
However, according to Italian daily Il Mattino, Napoli have decided that they will not sell the centre-back this month, despite the risk that means running.
Coach Gennaro Gattuso has made clear that he is depending on Maksimovic and does not want to see him sold.
Instead Napoli will look to find an agreement with the defender to extend his contract before the end of the season, to avoid him exiting on a free transfer.
Gattuso has handed Maksimovic the full 90 minutes in each of Napoli’s last three Serie A games.
He has made a total of 83 appearances for Napoli since joining the club in 2017 and played under current Everton boss Ancelotti.