The Colombia international starred for his country at the World Cup in Brazil, finishing as the tournament's Golden Boot winner with six goals from five matches.
His displays at the finals prompted Real to part with a reported €80 million to prise him away from Monaco, and he will link up with fellow costly captures Cristiano Ronaldo and Gareth Bale at the Santiago Bernabeu.
Butragueno has backed the 23-year-old to handle the expectation brought about by such a large fee.
"He is young and Real Madrid is a very demanding club," said Butragueno.
"Our fans are very loyal but, at the same time, they expect your best. Every game is like a test, so you have to play well. We have to win.
"So, it's part of our culture, our history. We have to win a title. It's what our fans expect.
"But he is a nice guy. I had the opportunity to meet with him during the presentation, and I think he's down to earth.
"He's looking forward to starting the season, to helping us. Then we'll see.
"But, really, we are very optimistic about what [role] he can play for us. We think he's very talented, a very gifted player."
Alessandro Del Piero, who called in on the Real squad at their training camp in Los Angeles, believes the club have recruited well in the off-season, with Toni Kroos also arriving from Bayern Munich.
"They reinforced the squad well," he said. "James is an incredible player - he's young and has a great future ahead of him; and Kroos will fit it perfectly - he is a midfielder who will play an important role for Real Madrid."
