The newly-promoted side have failed to persuade the 23-year-old target-man to join them from Liverpool in a £2million season-long loan deal.
Carroll does not appear to be central to Brendan Rodgers’ plans at Anfield and that had opened up the possibility of a switch but the player himself is thought to be unwilling to make the move.
Allardyce is disappointed with that decision – as he believes the arrival of the England international could have ensured a successful season.
“I would have probably stuck my neck out and said we would almost certainly stay up with him on board,” Allardyce told The People.
“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.”
Hammers skipper Kevin Nolan is hoping that situation changes as he would welcome the opportunity to link up with his former Newcastle team-mate once again.
“I would be delighted if Andy would come and join us here,” Nolan told the Daily Star.
“I am not too sure whether Brendan Rodgers will let him leave but there is lots of speculation – there always is surrounding Andy.
“If there is a chance, we would be in the reckoning for him because he is such a quality player.”