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.
"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."
Rafael Benitez's side have laboured in midfield since the sale of Alonso, with Lucas Leiva and Alberto Aquilani yet to convince supporters.
