The Spaniard's current deal at Bernabeu is due to expire at the end of June, and he is believed to be preparing to leave the European champions in search of a new challenge before the end of his career.
However, according to Marca, Ancelotti is fighting to keep hold of Nacho, with the Italian allegedly telling him: "Don't do something crazy, stay here, with us, at Real Madrid."
The report claims that Nacho's mind is not made up, and a one-year contract extension to take him to the end of the 2024-25 campaign has not been ruled out at this sage.
The 34-year-old made 45 appearances for Real Madrid during the 2023-24 campaign, including the full 90 minutes of the Champions League final despite Eder Militao being available for selection.
Ancelotti 'fighting' to convince Nacho to stay
Militao picked up an ACL injury in the team's opening match of 2023-24, which saw him miss the majority of the campaign, and Ancelotti used him sparingly towards the end of the season.
David Alaba, meanwhile, also suffered an ACL injury last December, which led to Nacho enjoying an extended run alongside Antonio Rudiger in the middle of the defence.
Nacho has been with Real Madrid since 2001, progressing to first-team level in 2011, and he has represented the capital giants on 364 occasions in all competitions.
The defender has won four La Liga trophies, two Copa del Rey titles and six Champions League crowns during a hugely-successful period in the Spanish capital.
Nacho is one of four Real Madrid players out of contract on June 30
Toni Kroos is retiring from professional football this summer, with the Germany international set to represent his country at Euro 2024 before hanging up his boots.
Ancelotti said after the Champions League final that he was hoping Kroos would reverse his decision, but the midfielder has almost certainly played his last game for Los Blancos.
Lucas Vazquez, though, is widely expected to sign a new deal, while the same can also be said for Luka Modric, with the Croatian preparing to pen a one-year extension.
Real Madrid could therefore potentially keep hold of Nacho, Vazquez and Modric for one more season, while the club have decisions to make over loan duo Kepa Arrizabalaga and Joselu.
Arrizabalaga is almost certain to head back to parent club Chelsea, having dropped to Real Madrid's number three goalkeeper after Thibaut Courtois's return from injury, but Joselu scored 17 times during a strong campaign and could now join on a permanent basis from Espanyol.