The Liverpool centre-forward has been on loan at Upton Park all season and after a slow start, became one of the stars of the Hammers line-up, prompting Allardyce to speak highly of his performances.
After the 4-2 win over Reading on Sunday to finish the season, Allardyce said he anticipated that Carroll had not played his final game for the East London club.
"I hope not," Allardyce said.
"I think that we all know we would like to secure his signature. We have got to sort that out with Liverpool and hopefully that becomes fruitful.
"The fans have made it plain and clear what they would like to see with Andy."