Talks between the clubs are at an advanced stage and Redknapp is increasingly confident of landing the 33-year-old former Manchester United defender.
It comes hot on the heels of Redknapp's attempts to bring Neville's former United team-mate David Beckham in on loan from LA Galaxy.
Enlarge Leader of men: Phil Neville offers Harry Redknapp a wealth of experience
Neville has 18 months of a four-year contract to run and, because of his age, the offer from Tottenham only extends that far. But Neville sees this as one last opportunity to play Champions League football, as well as to be part of the exciting team Redknapp has built.
A fee of around £1.5million has been proposed but Everton manager David Moyes, while reluctant to see his captain leave, might be prepared to consider a player exchange.
Roman Pavlyuchenko and David Bentley had been considered, although winger Bentley was yesterday at Birmingham City for a medical.
Loan target: David Beckham arrived at Tottenham on Monday to start training
Redknapp has the support of chairman Daniel Levy in signing Neville, who offers the versatility of being able to play at full-back or as a defensive midfielder.
On Tuesday morning, Redknapp said: 'He's a great player but I'm sure David Moyes knows just how important he is to his team. He's a real leader for them and David wouldn't want to lose him so it'd be a difficult one.'
But he has been asked to off-load one or two players before bringing in the former England man on a £50,000-a-week deal. Croatia defender Vedran Corluka, linked with a move to Liverpool, could be one to leave White Hart Lane.
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