The East Londoners have been linked with a host of managers from home and abroad with ex-Besiktas boss and Hammers cult hero Slaven Bilic the favourite to replace Big Sam at Upton Park.
But young tactician Eddie Howe, whose stock has gone through the roof after he led the Cherries to their fairytale promotion to the top flight this season, is another name consistently linked with the role.
Over the weekend, Howe briefly moved up to second favourite in the SkyBet odds to take up the vacant hotseat at the Boleyn.
But former West Ham chief Redknapp, who has himself been linked with a shock return to the club he served 476 times as a player and manager, has poured cold water on any talk of Howe being lured away from the coast.
"I was at the LMA (League Managers' Association) dinner and there's some good lads (around)," Redknapp told TalkSPORT.
"They're (clubs looking for managers) not going to get Eddie Howe out of Bournemouth.
"Garry Monk has done amazing and stepped up through the ranks at Swansea. There some good guys around."
But Redknapp says West Ham co-owners David Sullivan and David Gold, with whom he enjoys a close relationship, have their work cut out to replace Allardyce.
He said: "Sam did an excellent job there when you look at his record.
“It’s not going to be easy to replace him. Getting rid of somebody like Sam, to find someone to come in now and take the club forward will be hard.
“They’ve got great support and it’s a great opportunity for someone to go and take the club on and carry them forward.”
The 68-year-old Poplar-born boss was expected to announce his retirement when he walked out of QPR earlier this season, but he has since insisted he would be back in management at the earliest opportunity.
Redknapp had previously ruled out a return to Upton Park, but speaking to Sky Sports' Fantasy Football Club last month he admitted: "The club is in my blood... I couldn't sit here and say I wouldn't want to manage West Ham again because anybody who has played there would love to go back there, it's a great club."