Speculation has been rife that the 23-year-old could be set to leave Anfield after manager Brendan Rodgers conceded he would allow the forward to go out on loan.
Carroll has struggled to find form on Merseyside following his £35 million switch from Newcastle in January of last year, and a return to his former club and Fulham have both been touted.
However, after co-owner David Gold confirmed the Hammers intentions to bring in the England international on a short-term deal earlier this week, a move for the player looks increasingly likely – providing the striker is willing to join the Premier League new boys.
"You can't say you wouldn't be interested in Andy Carroll, he's a player who has got great stature in the Premier League," Allardyce told Sky Sports.
"Whether Andy Carroll wants to come to West Ham is another matter."