The Colombia international completed a move to the Santiago Bernabeu earlier this month in a deal reported to be worth around €80 million, after impressing at the World Cup in Brazil.
Having spent a season with Monaco in Ligue 1, Rodriguez is now preparing to bolster a squad already featuring formidable attacking talent in the form of Cristiano Ronaldo, Gareth Bale, Karim Benzema and Angel di Maria.
The former Porto playmaker does not consider himself a guaranteed starter, despite his price tag, and is ready to fight for a place under Ancelotti.
"I'm happy because being here is a dream come true. Hopefully everything goes well. I want to win titles at this great club," he told Caracol Radio.
"I hope I'm a starter but I have to earn my place. There are players who have spent years here and I have to respect that.
"I'll play where the coach wants. I can play on the right or left wing or as an attacking midfielder. I'll try to earn my place alongside these great players I have by my side.
"Wearing the number 10 is massive. It's something that with it carries a lot of pressure. I hope I can carry this number responsibly."
Rodriguez was also questioned as to whether he would welcome another world-class striker in Real's ranks - namely former Porto and Monaco team-mate Radamel Falcao.
"I'm sure he can play here. But I'm not the one who decides. It's up to the president," he added.
"I've been playing with him for four years. The only break was when he went to Atletico [Madrid]. He has a winning mentality and that's what you want at [Real] Madrid. Hopefully he can come."
