Grant was sacked after Sunday's 3-2 injury-time defeat at Wigan, a result that confirmed the Hammers' relegation from the Barclays Premier League.
Among the early frontrunners are QPR manager Neil Warnock, Cardiff boss Dave Jones and Martin Allen of Notts County.
Search: David Gold and vice-chairman Karren Brady need a new West Ham manager
Martin O'Neill, who came close to replacing Grant in January, could re-emerge as another contender.
Gold told Sky Sports News: 'The sooner the better because we have a lot of work to do - an enormous amount of work - but we mustn't be rushed.
'This will be a very, very importantappointment. We're choosing a manager good enough to get us out of thisdivision and capable of continuing once we've achieved that.
In the frame: QPR boss Warnock
In with a shout: Cardiff's Jones
Shortlist: Notts County chief Allen
NEXT WEST HAMMANAGER ODDS Steve McClaren - 8/15
Neil Warnock - 9/1
Chris Hughton - 9/1
Sam Allardyce - 12/1
Gus Poyet - 14/1
Martin O'Neill - 16/1
Paolo Di Canio - 16/1
Dave Jones - 33/1
Martin Allen - on request
Odds courtesy of Paddy Power
'Now relegation is known this is oneof the reasons we moved quickly. It's important we have time for the CVs to come into us. I'm sure many, many managers will see the job as challenging and exciting and it's important we pick the right person.
'We're also prepared for the most exciting venture in the history of the club in the sense that we'll be moving to a new stadium, which is something we've never done. All these challenges have got to be addressed, so it's important we pick theright person.'
Gold is scathing in his analysis of Grant, giving credence to the belief the Israeli struggled to inspire his players.
He said: 'Everything seemed OK to me when I visited the training ground, but my visits are quite short otherwise I get accused of interfering and that's the last thing I'd want to be accused of.
'I'd be in the dressing room before amatch and the training ground for half a day and what I saw was professionalism - but that's only what I saw.'
The departure of West Ham's top players is inevitable, yet Gold will not let them leave without a fight.
Scott Parker and Rob Green will lead the exodus, while the contracts of Matthew Upson, Kieron Dyer, Danny Gabbidon and Lars Jacobsen expire this season. On-loan Wayne Bridge and Robbie Keane will return to their clubs as a summer of upheaval at Upton Park beckons.
Axed: Avram Grant was sacked after West Ham were the first side to be relegated to the Championship this season
HAVE YOUR SAY. Who do you think is to blame for West Ham's slump? West Ham will play Championship football next season after they suffered an agonising defeat at relegation rivals Wigan on Sunday. Manager Avram Grant has already been sacked by the board, while Scott Parker will lead the exodus. But who do you think is responsible? TELL US WHAT YOU THINK 'We will do everything in our power to keep our best players,' said Gold.'It will be very difficult to ask an international to come and play in the Championship when traditionally England players aren't picked from the Championship.
'By the same token I'd hasten to add that we're not in any hurry to sell. We must receive the proper amount of compensation for any player we sell and if we don't then that player will stay with us in the Championship.'
Gold insists the 'grieving' process over West Ham's relegation must soon give way to a determination to secure an immediate return to the Premier League.
'We must prepare a squad of players to get us back into the Premier League,' he said. 'West Ham is a great club with a great tradition. Our fans are all hurting and feeling sorry for ourselves now and so am I.
'We're entitled to grieve and that can go on for a couple of days or maybe a couple of weeks, but sooner orlater we have to stop feeling sorry for ourselves and start talking about promotion.
'We've got to do everything in our power to get back into the Premier League as soon as possible.'
Hammer blow: West Ham's players can't hide their disappointment after failing to beat the drop
Decision time: Co-owners David Sullivan and David Gold must choose a successor to Avram Grant in the Upton Park hot seat
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Explore more:People: Martin O'Neill, Matthew Upson, Neil Warnock, Chris Hughton, Robbie Keane, Lars Jacobsen, Danny Gabbidon, Sam Allardyce, Kieron Dyer, Wayne Bridge, Martin Allen, Steve McClaren Places: Cardiff, United Kingdom