Redknapp has replaced Alan Shearer as the odds-on favourite to be installed as the next manager of Newcastle and Hills have slashed the Pompey boss to just 2/7 to take over.
Local legend Shearer was initially chalked up as the 4/6 favourite by Sky Bet just moments after the shock news broke that Allardyce had parted company with the St James' Park club.
Initially a 20/1 chance at Hills, Redknapp's odds were quickly slashed and less than three hours after the market had opened, he was odds-on favourite.
Mark Hughes was the other British manager punters latched on to and the Blackburn boss was slashed to 7/2 from 25s by bet365 before settling back at 9/2.
Shearer's chances of getting the job seemed to take another blow with the BBC reporting on Wednesday evening that sources close to the former England striker - who works for the corporation as a pundit - were saying he was extremely unlikely to succeed Allardyce at Newcastle.
And further reports on Sky Sports News that the club were looking for an "experienced" manager spelt more bad news for Shearer backers and prompted a further cut of Redknapp to just 2/7 with Hills.