Bookmakers slashed the odds on Redknapp taking the hot seat at Stamford Bridge after a surge of three-figure bets were placed on Tottenham manager on Thursday.
Redknapp insists he is happy at his current club but the fluent attacking football he has brought to Tottenham, and his success in the Champions League this year, make him a plausible target for Roman Abramovich, who is looking for a new manager after sacking Carlo Ancelotti last week.
Welsh wizard: Gareth Bale is confident Tottenham will get back into the top four
Redknapp has succeeded where Martin Jol and Juande Ramos failed in getting the best out of Bale, who has become one of the most sought-after players on the planet thanks to a series of stunning displays this year, most notably the two performances against Inter Milan in the Champions League.
The 21-year-old says Redknapp has been key to his and the club's progression, and he therefore wants him to remain.
Better together: Bale and Redknapp enjoyed a stellar campaign in Europe this season
'I want him to stay, definitely,' Bale said. 'Since I've been at the club we've had three managers and it's obviously nice to have him there because he has brought a bit of stability.
'We have got used to him and how he manages and how he wants us to play so it will be great to keep him. Nobody wants him to go and the club, I'm sure, will do everything they can to keep him.'
Bale's own future has also been a source of constant speculation throughout the year.
The Welshman signed a new contract in March, but rumours persist of big-money bids from the likes of Manchester City, AC Milan and Inter this summer.
Bale admits that playing abroad would one day appeal to him, with Barcelona being one possible destination, but he insists he is happy at Tottenham for now.
'I left home when I was 15 and it was quite difficult then but if I have to move abroad then it's something I wouldn't be scared of,' Bale said. 'Barcelona play great, great football but I'm happy where I am.
'I'm focused on what we want to achieve as a team next season and I'm enjoying my football here. I'm still 21, I'm getting a good run in the team and I'm still learning my football.'
Gareth Bale is a Sony 3D World ambassador and was speaking from the Sony Stadium part of the UEFA Champions Festival.
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Explore more:People: Daniel Levy, Harry Redknapp, Martin Jol, Roman Abramovich, Juande Ramos, Gareth Bale, Carlo Ancelotti Places: Barcelona