The Blancos are keen on securing the Tottenham star at potentially a world-record price and his Wales boss feels that he would cope with the pressure
Negotiations have begun between the two clubs over a potential deal, though nothing has yet been agreed, but the international manager is confident that the Spurs star would not be overawed by a switch to the Spanish spotlight.
"He can stop and think 'wow, look at the situation I'm in' but he's not the type of lad to do that," Coleman told Sky Sports News.
"He's unassuming, he's quiet and you don't get to the level he's got to by having the mentality that you can just rest on what you've achieved already.
"The best players, the reason they're as good as they are, it's not just because of their talent - it's because of their mentality. They can handle pressure-cooker situations and they just take it in their stride. Gareth's got that mentality where he doesn't get carried away.
"Time and time again he scores a goal and you think 'how can he top that?' And he does with something else. He can take your breath away. He does that consistently."
Coleman previously coached La Liga outfit Real Sociedad and he is aware of the importance of Bale adapting to life in Spain if he joins Carlo Ancelotti's side.
"If there is a big transfer, he's not the type of player to get carried away with it," the Wales boss added. "If it does happen - I'm not saying it will, but if it does - he'll have to learn the culture quickly. That's what I found out when I was in Spain.
"Not everybody changes for you because you're from a different place. You've got to get in the middle of it. Learn the language, learn the culture and embrace it, I'm sure he'll do that."