The Dutch midfielder has started just five games in La Liga this season and was reported to be seeking a move away from Spain in order to secure regular action ahead of this summer's World Cup, with both the Reds and the Gunners believed to be keen.
Former Madrid coach Bernd Schuster signed Van der Vaart from Hamburg for €13 million in 2008.
He has found first-team opportuntiies harder to come by following the arrivals of Kaka, Cristiano Ronaldo and Xabi Alonso.
However, he has insisted that he still believes he is good enough to play for Los Blancos and is not interested in a move abroad to either Anfield or the Emirates Stadium.
"I have the skill and quality to play for Real Madrid," he told Radio Marca.
"Pellegrini told me in pre-season training that I was a very good player, but he did not have much room for me because they had Kaka, Raul, many players. I said OK, but I will not leave Madrid, I want to fight, so let's see.
"The league is very important but also the Champions League. The final is in Madrid. It is a dream for everyone to play the final here, but we also want to win the league.
"I am very happy in this team."
Arsenal had been tipped to make a move for the Dutch international with both Cesc Fabregas, Denilson and Tomas Rosicky struggling for fitness, while Alex Song is currently away with Cameroon at the Africa Cup of Nations.
Rafael Benitez's side have laboured in midfield since the sale of Alonso, with Lucas Leiva and Alberto Aquilani yet to convince supporters.
Liverpool did, however, seal the signing of Atletico Madrid winger Maxi Rodriguez on Wednesday.
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