The Reds visit Upton Park fighting it out with Manchester City and Chelsea to be crowned Barclays Premier League champions with both Carroll and Downing deemed surplus to requirements by Brendan Rodgers since he took charge at Anfield.
The England international duo were snapped up by Allardyce, with Joe Cole also returning to West Ham from Liverpool in January 2013, and the 59-year-old knows that those players will have to put in top-drawer performances if they are to hurt the title tilt of Rodgers' side and make their point to the Northern Irishman.
"He (Brendan Rodgers) made a decision early on (with Carroll) and we were the first to try and resolve that situation," said Allardyce.
"As disappointed as Andy was, that was the decision made by Brendan and you have to move on, you have to put it behind you.
"Stewart Downing also comes into the fray where he played a part in Liverpool last season and then joined us. The rest of the team and those two lads particularly will want to show how good they are on Sunday. Let's hope they play very, very well on Sunday because we need them to."
With Luis Suarez and Daniel Sturridge forming a seemingly free-scoring strike partnership for the visitors, Allardyce will have to be wary of the goal threat but, as a proud Englishman, he is also happy to see a number of homegrown players emerging.
Along with Sturridge and captain Steven Gerrard, England manager Roy Hodgson named Glen Johnson, Jordan Henderson and Raheem Sterling in his side against Denmark last month as half of his outfield team was made up of Liverpool players.
"People are emerging from an England point of view and it's very encouraging to see Jordan Henderson and Raheem Sterling emerge more than ever this year," Allardyce added.
"We've got a big job on our hands, we know that, but after our performances and our results we're really looking forward to it.
"We've eased the pressure on ourselves and produced a great run of form and results at this period of time with six wins out of nine. That's two points per game, that's top-six form. We're in our best form of the season and hopefully that will make for a competitive game on Sunday afternoon."
Allardyce has confirmed captain Kevin Nolan will be available for the game having been forced off late in the win over Sunderland.
The 2-1 victory over the Black Cats was West Ham's sixth win in nine games and, after a spell in the bottom three over the turn of the year, Nolan is now targeting back-to-back top-half finishes.
"It's nice to be sitting where we are now with six games to go and we're looking forward to what hopefully is going to be a few more wins," he said.
"Our aim is to knock Stoke out of that 10th spot and finish where we did last year.
"Then we can all go and have a good summer and let the gaffer and the board start worrying about who they are going to bring in for next season - we're not far off being able to celebrate come the end of the season at another successful campaign.
"Monday's victory at Sunderland was a massive one for us. It gave us another set of back-to-back wins and took us to 37 points, eleven ahead of the relegation zone and just three behind Stoke in 10th.
"It put a smile on everyone's face and lets us think we've got another season in the Premier League when we won't have as many major injuries as we had this season."