The midfielder succeeds Nacho Fernandez after his departure for Al Qadsiah.
The 38 year-old signed a new 12 month deal this week and said, “It's an enormous responsibility. When you look back at the captains Real Madrid has had over the years, to get the chance to be Real Madrid captain is beautiful, it's so special and I'm thrilled to represent Madrid as its captain.
“I never dreamed of everything that has happened to me, especially having spent so many years here. Genetics comes into it, and I'm lucky in that sense, but training here with so many young lads who push you to work hard, that's what makes you better every day. You can never afford to relax. Training outside Valdebebas as well, I look after myself and do all I can to be at the right level at my age."
Modric also said: “Having the chance to become the player with the most trophies in Real Madrid history is unbelievable. There are so many titles to be won by this team and the excitement is there, both for me and the rest of the team. I'm ready for what's to come. I hope we can win a lot this season and I can become the club's most decorated player."
On Kylian Mbappe's arrival, he added: "He's one of the very best and he's shown it over the years. He's a player who has won the World Cup, against me, unfortunately, but I'm delighted to have him on my team now. We'll all help him out with whatever he needs to make sure he settles in quickly and shows his talent, because he can reach the very top."