The 24-year-old is working alongside the Portuguese at the Santiago Bernabeu after completing an £86 million transfer in the summer from Tottenham.
For his former manager at White Hart Lane, the psychological test of playing alongside Ronaldo will prove most important to Bale’s success or failure.
“His relationship with Ronaldo is the key to it all, because if the football is going well then all the added stresses are a minor irritation and nothing more," Redknapp wrote in his new book, being serialised in the Daily Mail.
“His biggest test will be to step out of the shadow of Cristiano Ronaldo with confidence. That won’t be easy
“Ronaldo is a huge star at Madrid and will probably want to take nine out of 10 free-kicks - at least. Gareth will have to assert himself and that will require a strong mind. He has to think ‘I’m an £86million player’ and act like it, taking responsibility, claiming the ball when he fancies his chances...
"The one thing the club cannot provide for Gareth and Cristiano is a ball each - so they will need to work hard on that partnership because they are such similar players."
Redknapp fears that Bale might be overawed and intimidated by Ronaldo, since the Welshman is not naturally assertive.
"When he goes it alone, has a shot, misses and Ronaldo starts throwing his arms up in the air... He cannot, at that point, go into his shell and become this timid little creature. But it is not natural for Gareth to behave in an assertive way."
The former Tottenham boss added that Bale does not suffer from self-doubt, however, merely that he is unassuming.
"Don’t get me wrong, he knows he is good... He wouldn’t have fought so hard to get the deal done if he didn’t fancy his chances of living up to expectations in Madrid. Yet, equally, Gareth is a quiet lad, who spends time with his girlfriend and family. I’m not sure being in the same bracket as Ronaldo and Lionel Messi will suit him...
"He had a little trouble adjusting to the bigger environment of Tottenham after leaving Southampton and this is 10 times as great as that move. If I have a worry it is that I remember the days when Gareth’s confidence was draining fast at Spurs and there were genuine fears he might not make it... He won’t like the attention a difficult start brings either. Gareth is a very private person and he won’t enjoy having every move scrutinised"
Redknapp added that Bale will need to make the most of the familiar faces at the Bernabeu: "He will need to lean a lot on [former Chelsea manager Carlo Ancelotti], who speaks good English, and Paul Clement, Carlo’s assistant, who is English. Luka Modric is another old friend who could help him settle in."