"He's as I expected him to be. He is the first to arrive at a training session, he has fully integrated into the squad, he doesn't act as a star," he said during his first post-training press conference as Real's new coach.
"He is a very talented but normal player who is devoted to Real Madrid. His image is very different to how he is as a player."
Real signed the player from Manchester United last month on a six-year deal worth around 94 million euros (£80 million) and he will reportedly be paid 13 million euros each season, making him the most expensive player in the world.
The 24-year-old was officially unveiled before some 75,000 ecstatic Real fans on July 6 at the Santiago Bernabeu stadium.
He had to be hurried away when spectators leapt over security barriers to try to get an autograpah.
Real has spent over 200 million euros in the pre-season to sign new talent, including Brazilian playmaker Kaka and French international striker Karim Benzema.
Pellegrini said he hopes the squad will be completed by the end of the month.
"The sooner the better. July 27 would be good because that's when all our internationals return," he said when asked for an ideal date to have his squad finalised.
"But I'm not bothered about waiting a while longer if there are any doubts concerning a signing. There's no deadline. I know what I want, but I'm not looking to have it for our first pre-season match," he added.
