The Hammers are trying to make the Olympic Stadium their own after the conclusion of the Paralympic Games in September.
West Ham were initially denied tenancy of the stadium that hosted the Olympic Games last October following a legal dispute involving Tottenham Hotspur and Leyton Orient.
Allardyce insists that they should continue to strive to move to the Stratford venue for future growth and prosperity.
"When you see the atmosphere created in that arena then I think you would dream of walking out there, like you did at Wembley," he explained. "Every footballer's dream is to walk out at Wembley and I think when you've got a venue of that size [it would be the same].
"All the players look forward to playing at Old Trafford and at the Emirates. Not quite so much these days at Liverpool or Everton because they're old stadiums now."
"But Manchester United is now almost completely redeveloped and has 75,000, while the Emirates is one hell of a place when it is full. We could be that size in a brand new stadium, with that atmosphere.
"It would be awesome to walk a team out on that pitch and say 'this is West Ham's new home and the creation of what could possibly be a new modern history hopefully'."
West Ham currently play their home games at Upton Park, which can host 35,000 spectators.