The 52-year-old has been impressed with the Welshman following his move to los Blancos and believes he and Ronaldo will be a force to be reckoned with in La Liga this season
The 24-year-old spoke a sentence in Spanish in his first press conference as a Madrid player, impressing the media and fans alike, and Lineker believes it is crucial for the Wales international to embrace life in Spain away from the pitch and learn the language.
"It makes all the difference. He didn’t just do two or three words, he uttered a whole sentence. He had probably been given a bit of guidance but good for him for having a go and the accent was fine," Lineker told Absolute Radio's Rock 'N' Roll Football.
"It's appreciated...especially in a place like Real Madrid, big clubs like that, where they understand you are immediately making an effort to integrate yourself.
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"When I went to Barcelona, I looked at all the players that had been pretty successful abroad, Laurie Cunningham, Ray Wilkins, [David] Platty, Trevor Francis, Steve Archibald – the ones that did really well, where the ones that sort of immersed themselves in the culture and learned the language etc and I don't think it's any coincidence."
The former England international is confident Bale and new Madrid Team-mate Cristiano Ronaldo will work well together, and is excited by the Blancos duo going head-to-head with Barcelona pair Lionel Messi and Neymar.
"I think they might score a few goals," Lineker continued. "The problem is where they can balance it at the other end but I mean, what a team going forward that will be. Ronaldo is something else. Bale has massive, massive potential; he was unbelievably good last season.
"It might take him a while to adapt, shame he's had one or two niggly injuries going into it, but prodigious talent, so they will certainly be worth watching and then you have got Messi and Naymar at the other place, so not bad."
Bale left Tottenham to move to the Spanish capital, and Lineker believes his former club have reinvested the money from the Welshman's sale wisely and can finish ahead of north London rivals Arsenal this term.
"They will probably finish above Arsenal," Lineker added. "The interesting thing is that normally clubs, if they sell a huge player, they always say, 'oh, we'll spend the money, we will put it back in the team' but they invariably do not.
"Whereas at least you can say for Spurs they've had a plan and they've spent that money – and they spent it before they got it – so they have bought pretty well on the whole. Even Arsenal are having a go in the transfer market these days – what is going on?"