The nine-time European champions announced Bale's signing from Tottenham on Sunday evening for a fee understood to be 100million euros (£85.3million) with the Welshman agreeing a six-year deal.
He becomes the latest in a string of Britons to move to Spain, following Steve McManaman, Michael Owen, David Beckham and Jonathan Woodgate to the capital club in the last 20 years.
Lineker joined their Catalan rivals from Everton and enjoyed three successful years in Spain before moving back to England to join Tottenham.
And he is sure Bale has all the attributes on the field to thrive in the Primera Liga, but admits that the 24-year-old will have to cope with a new kind of pressure on the continent.
He told Radio Five Live: "It's whether he's got the mental strength to be able to cope with such a big move and the spotlight that will be immense on him over there in Spain.
"Because it is different, it's huge and it's whether he can cope with that.
"Ability wise he is phenomenal."
Lineker has also advised Bale that getting to grips with the local language should be a number one priority for his European adventure.
"Wish @GarethBale11 all the very best with @realmadriden. A fabulous opportunity that any top player would seize," he said on his Twitter account.
"From someone who did something similar eons ago, one piece of advice: Learn Spanish. Communication is vital. His football will do the rest".
Mark Hughes played alongside Lineker at the Nou Camp and he has also urged his fellow countryman to embrace the foreign lifestyle.
Speaking before the move was completed, he said: "I would just tell him go and embrace everything about it.
"I probably went with the wrong mentality in terms of thinking 'I'll go out here, have a couple of years, and then I'll get back'.
"That is totally the wrong way to go about it. You need to just embrace everything - the language, the lifestyle - and if you do that, it makes what you do on the football pitch flow a lot easier."
He added: "When I went there I was a young player, but I was a young man as well and I hadn't matured to the level I needed to to be able to cope with going from Manchester United to Barcelona. It was probably too soon for me.
"But if he feels confident enough that he's able to cope with the focus that will be on him, then I wish him well."
Owen spent a single season at the Bernabeu after signing from Liverpool in 2004 and he believes the forward will be playing "on the biggest stage of all".
"What a deal all round," he Tweeted. "Spurs rake in a fortune, Madrid get a World Class player and Bale gets to ply his trade on the biggest stage of all."