Swansea City splashed out a club-record £12 million on Bony from Eredivisie outfit Vitesse Arnhem in 2013, the same year West Ham brought Carroll back to Upton Park on a permanent basis.
The two Premier League rivals meet at Upton Park on Sunday and Allardyce said he made enquires about the Ivory Coast international before spending £15m on Carroll, who netted seven goals in 24 matches during his loan spell from Liverpool in 2012-13.
Bony has gone on to score 22 goals in 48 Premier League appearances, while injuries have limited English striker Carroll to a mere two goals in 19 games but Allardyce still believes he got the better deal.
"Hopefully, in the end, it will be an Andy that is not picking up injuries and starts getting back to what we know he can do," Allardyce told reporters.
"His first season in the Premier League at Newcastle he scored 11 goals in 16 or 17 matches.
"That showed at that young age he was a massive goal-scoring threat against anyone. He has done it before. Because of the injuries he had, that career hasn't blossomed as it should have done.
"It was one [player] or the other, it couldn't be both. We got the deal done with Andy and we couldn't do both - we didn't have enough money in the budget that year."
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