The Colombia international sealed a "dream" move to Los Blancos from Monaco earlier in July on the back of his Golden Boot-winning exploits at this summer's World Cup in Brazil.
Consequently, expectations are high among Madrid fans but Butragueno believes that the attacking midfielder has the right mentality to deal with the pressure of proving his worth to the club's faithful.
"He is young and Real Madrid is a very demanding club," Butragueno told reporters.
"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 [James] 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."
Rodriguez claimed the Golden Boot after netting six goals in just five games for Colombia at Brazil 2014.