Concise News reports that Belgium star, 28, joined from Chelsea last week for a fee that could exceed £150m.
After Hazard underwent a medical and then signed his five-year deal beside Real president Florentino Perez.
He said, “my dream ever since I was a child has been to play for Real Madrid.”
After signing the contract, Perez introduced Hazard, whose first speech was in Spanish saying: “Hola todos” (hello everyone), before continuing with a short speech in French.
He then changed into his Real kit and stepped onto the pitch, joggled a gold and white ball, kicked the ball into the crowd and later kissed the club’s badge.
In a news conference, he said: “I’m going to try to become the best player in the world, but first I wanted to be in the best team in the world.
“I’m not a Galactico, but I’ll try to be. For now, I’m Eden Hazard.”
Hazard is yet to have a number on his shirt, as the number 10 shirt he wore at Chelsea and for Belgium is worn by Luka Modric at Real.
“I’ve been lucky enough to talk to Modric through Kovacic, I jokingly asked him to leave me number 10 and he said no,” Hazard said adding that, “The number for me is not important.”
Hazard also disclosed that Real had made a move to sign him after last year World Cup, but said now was “the best time” to join the Spanish giants.
“I don’t think it is too late, I am 28 years old, the best moment for a player,” he said.
Zinedine Zidane returned to the club for his second term as a manager in March and Hazard said the Frenchman was a factor for being in Real.
“Everyone knows Zidane was my idol growing up,” Hazard said. “That he is the coach has been an important factor but not the only one. I already wanted to play here.”