The Hammers boss admitted he would like to keep the 24-year-old at the club beyond his current loan spell and also revealed that George McCartney has been ruled out for the season
Carroll has netted five times in 17 Premier League appearances while on loan from Liverpool this season, including a double during their 3-1 win over West Brom on Saturday.
And Allardyce has admitted that a full transfer is "always a possibility", although the club would have to wait until the end of the season for any agreement to be reached.
"Yes, I would like him to stay but there’s a lot of things that need to be done to ensure this transfer," the 58-year-old told reporters. "We’d have to wait and see until the end of the season.
"There’s always a possibility of having him here full-time. When we get down to budgeting and negotiations, we’ll have a look at anything that comes our way."
Allardyce was delighted with his side's victory, which lifts them to 11th in the Premier League, but revealed that spirits were dampened by news of a re-occurence of George McCartney's medial ligament injury, which rules him out for the rest of the season.
He explained: "The only down side to today was that George McCartney has got injured again.
"I would have thought [that he is out for the season], it’s a re-occurrence of the same injury, which is a big disappointment for him and for us.
"We’re disappointed for George as he’s been so eager to come back and with the little run out we gave him, sadly he’s gone down with an injury again."
West Brom manager Steve Clarke, meanwhile, was left frustrated by the nature of his side's defeat and spoke in particular about Youssouf Mulumbu - who was sent off late on for kicking the ball at Gary O'Neil in frustration.
"I just said after the game to the group in general that it was a frustrating afternoon but you have to control your frustration," Clarke said.
"You have to control yourself because the red card doesn’t affect today’s game but will now affect us over the next three games – we have a small squad. I will have a chat with him tomorrow morning."