The 29-year-old arrived at Goodison Park while Ancelotti was at the helm and, during his debut season as a Toffee, Rodriguez scored six goals and provided five assists in 23 Premier League appearances.
His relationship with the Italian manager has been developed at three different clubs as he previously played for the 62-year-old at Los Blancos and Bayern Munich.
When discussing his future, Rodriguez told reporters: "Return to Real Madrid [to play] under Ancelotti? I don't know, I don't think so. We'll see. If I stay another year at Everton, that's fine."
The Everton midfielder was snubbed from Colombia's Copa America squad for this summer and Rodriguez went on to add his disappointment: "I don't agree with the decision of the [Colombia] coaching staff because they disrespected me."