The Hammers are close to securing their Barclays Premier League future ahead of Sunday's trip to Carroll's parent club Liverpool.
Allardyce, who is optimistic an extension to his contract which currently expires at the end of the season is close, is keen to realise the ambition of the Hammers hierarchy to have a team worthy of the Olympic Stadium and turn the club into a Premier League force.
But the former Bolton boss admits football's new financial constraints might mean patience is required.
"There's a big motivation for everybody to try to get West Ham to become the biggest club it has ever been. That has to be the ultimate goal," Allardyce said.
"But the days of huge money-throwing transfers look like they might be restricted so the growth of West Ham might not be as quick as we would all like, certainly me.
"Everybody has to be very aware that we might not be as able to be free-spending as we would want to be to get the team the best as it could be as quick as we can."
Carroll scored twice against West Brom last weekend in a 3-1 victory which virtually assures safety.
The 24-year-old England striker moved to Liverpool for £35million in January 2011 but was sent out on a season-long loan to Upton Park and will be ineligible on Sunday.
Allardyce was sporting a plaster above his eye, covering four stitches, after slipping in his kitchen at home prior to a visit to the Grand National Festival at Aintree.
Sunday's return to Merseyside could be the first of a few trips to the north west as Allardyce aims to secure Carroll on a permanent basis for the Hammers.
For his personal development and England's sake, Carroll must play regularly, according to Allardyce.
"He has got to go to play in somebody's first team and score goals and continue his development," Allardyce said.
"The lack of goal-scoring players in the England squad now is very limited.
"If you look at the last England squad when they played in Montenegro there was only Wayne Rooney and perhaps Steven Gerrard from midfield. Danny Welbeck, as good a player as he is, has not been a goal-scorer.
"Andy hasn't proven to be a goalscorer this year but we have seen him score lots of goals in the Championship and for Newcastle in the Premier League and starting to score for us.
"I would be highly surprised if he didn't get back in there (the England squad) but to do that he needs to be in somebody's first team on a regular basis."
West Ham will wait until the end of the season to begin negotiations for Carroll in earnest and with Allardyce over an extended stay.
Former Hammers forward Paolo Di Canio was linked with Allardyce's post before taking control at Sunderland this week and the latter expects no problems in agreeing to a new deal.
"There would have been a lot more than Paolo Di Canio after my job, don't worry about that," he said.
"I'm not so sure the owners would succumb to appointing a fans' favourite.
"They are very wise in the world of football and if I wasn't going to be here they would look at everyone who was available to come to manage this great club.
"I'm just pleased that I don't see any problem that it it's going to be me."