The Hammers are hoping a lengthy application process will see them given the green light to move into the Stratford arena once the London Games come to a close.
The events which have unfolded in the stadium over the last few weeks, and are set to continue over the course of the Paralympic Games, have inspired a generation.
West Ham hope they can buy into that feel-good factor and inherit a world-class sporting venue.
An 80,000-seater stadium would allow them to bring in extra matchday revenue, offering a considerable boost to their bank balance and spending power.
In time, the Hammers feel they would then be able to stand alongside their more illustrious capital rivals.
"When you feel the atmosphere created in that arena, then you dream about walking out there the way you dreamed about walking out at Wembley when you were a kid," said Allardyce.
"Everyone looks forward to playing Arsenal at the Emirates. We could be the same size in that stadium.
"It would be awesome to walk a team out on that pitch."
