Carroll has been made available by Liverpool and West Ham were hoping to secure an ambitious deal for the England international.
Indeed, it was reported that the Hammers' offer of a £2 million loan fee for the season and a future fee of £17m had been accepted by Liverpool.
However, it is believed that Carroll, who cost Liverpool £35 million when joining from Newcastle in January 2011, will not consider joining a club that is not in European competition.
"I would probably have stuck my neck out and said we would almost certainly stay up with him on board," Allardyce said.
"We wouldn't have thought we would have had any chance of signing a player of Andy Carroll's capabilities but it became clear a couple of weeks ago there was a possibility of a loan. Unfortunately at this moment, it's not the case."
The West Ham manager was surprisingly sacked by Rovers owners Venky's in December 2010 and while he was leading his new club back into the Premier League last season, the Lancashire side were heading in the opposite direction under Steve Kean.
While the relegation of Bolton, another of his former clubs for whom he still has great affection, saddened Allardyce he was unconcerned by the demise of their north-west rivals.
"I have to say there wasn't much feeling when Blackburn were relegated because of the way they treated me," he told the People. "Bolton is a different story because I left them and I only did that because of their lack of ambition. Everyone knows how much the club means to me and always will, so I was saddened by their relegation."