Carroll, who moved from Newcastle United to Liverpool for 35 million pounds in January 2011, played only two games for the Reds at the start of last season before manager Brendan Rodgers loaned the striker to West Ham.
The deal was made permanent last May, with the London club paying 15 million pounds but Carroll was injured until January. Since then, the powerful striker may have scored only two goals in 10 games but he has provided a focal point for the attack.
Allardyce told reporters on Friday that had Carroll been fit all season, West Ham, who were in relegation trouble until recently, could have been challenging for a place in Europe.
“It’s a possibility, yes,” he said. “Since the turn of the year, our change of form and results has got better and better and now we’re in the best run we’ve had since we’ve been back in the Premier League.
“We’ve got to take that confidence into Sunday’s game and hopefully we can produce one of those performances that has happened many, many times in the Premier League this year.
“Just the other week Crystal Palace beat Chelsea, and Manchester City lost to Sunderland, things like that can happen.”
Former England winger Downing played 29 League games for Liverpool last season – including in their 3-2 win at West Ham – before moving to east London for a reported five million pounds fee in August.
“The rest of the team and those two lads particularly will want to show how good they are on Sunday,” Allardyce added. “Let’s hope they play very, very well, because we need them to.”
The West Ham manager praised Liverpool’s Luis Suarez, the Premier League’s leading scorer with 29 goals, as “one of the top two or three in the world” and described Sunday’s opponents as “in terms of form and results, operating at 100 percent”.
Liverpool won their eighth league game in a row last weekend to replace Chelsea at the top of the table and the Merseysiders will be crowned champions for the first time since 1990 if they win their six remaining matches.
(Writing by Steve Tongue; Editing by John O’Brien)