Alonso's value to Real's midfield has been more evident than ever this season with his leadership and ability to build play from deep, but he is in the final year of his contract at the Bernabeu and has so far stalled on signing a new deal.
The 32-year-old's financial demands are believed to be the sticking point and from Jan. 1 he will be free to negotiate with other clubs.
There are several Premier League sides thought to be interested in signing the former Liverpool man, although Real would be reluctant to lose such a key player.
"I don't need to convince anyone of the importance of him to the team," Ancelotti told a news conference.
"He has experience, quality and is one of the best midfielders in the world and that is why we are pushing for him to stay.
"All I can do is count on him until the summer and make the most of him while he is here. He gives more confidence to the team but we have other important players and we have also got good results without him."
Real go into the final match before the winter break looking to bounce back from a 2-2 draw away to Osasuna last weekend which left them five points off La Liga leaders Barcelona and Atletico Madrid.
On Sunday they play a Valencia side in crisis, having sacked coach Miroslav Djukic this week.
The Serb was dismissed after a poor start to the season, with the club reportedly in talks with Juan Antonio Pizzi of Argentine side San Lorenzo.
"This is a big game for us and we are up against a team who are changing coach so the players will be extra motivated to show what they can do. They are likely to be more motivated but we want to finish the year well, perform well and take home the points," said Ancelotti.
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