A unanimous decision from the London Legacy Development Corporation board (LLDC) was to make Premier League club the preferred bidder for the use of the Olympic Stadium.
Allardyce believes the relocation could be "outstanding" for the Hammers but has issued a stark warning that the club must continue to develop and progress to make the Olympic Stadium move work.
"There is one thing you have to bear in mind, you cannot build a white elephant and the white elephant is a great stadium and no team - disaster," he said.
"We have to manage our finances carefully because of the over-riding debt of the football club.
"I think what (co-chairmen) David Sullivan and David Gold do is top that up with their own wealth but there is somewhere down the line where they want to create a situation where that doesn't happen.
"The long-term future of the football club is not only a new stadium but a new training facility that needs to go with the stadium as well - there are years of good financial management needed as well as success on the field."
The 58-year-old Allardyce took charge at Bolton two years after they had opened their new stadium and soon led the Trotters to the Barclays Premier League.
The ex-Blackburn boss reckons his success in Bolton would not have been possible if the club had not moved to their newer, modern ground.
"There is always a lot of disruption when a club decides to move form somewhere which has a huge history," he said.
"It is always going to be upsetting to some of the very loyal supporters we have but in the end, when you look to what happened at places like Bolton, I couldn't have built that sort of success without those fantastic facilities - not that a facility has ever produced a player but it doesn't half help."
Allardyce is also aware that the extra revenue that comes from selling out a bigger stadium could make a difference in football's modern era.
"I suppose it has got bigger facilities and could then create bigger revenues and I think you have got to judge a football club by its revenues these days - by its spending power," he said.
"Its spending power is the ultimate now when we look at what Roman (Abramovich) has done at Chelsea and certainly what Mr. Mansour has done at Man City, so if you increase your spending power you can increase the size of your football club."
Sullivan and Gold have already backed Allardyce in the transfer window and also stumped up the wages to bring Andy Carroll to the club on loan.
The Liverpool striker has had some off-field distractions during his career to date but, following accusations of attacking a photographer in Dublin after the Hammers' Christmas party, Allardyce is not worried about Carroll's reputation.
"We have got 25 players out there and this is just a little minor blip for me that was caused by a photographer and not by Andy Carroll," he said.
"He just wants to get on with his football, he doesn't get distracted, he would prefer not to have it but like I said with the way he is he has got to cope with it as best as he possibly can."