A 2-0 defeat to title-chasing Leicester City leaves the Black Cats four points behind Alex Neil’s side in 18th place, leaving their top flight status hanging in the balance.
Having suffered a 1-0 defeat to Crystal Palace on Saturday, Norwich return to Carrow Road looking to secure back-to-back home wins for the second time this season, after edging past Newcastle last week.
While Sunderland have a game in hand over the Canaries, Allardyce says that defeat would be a hammer-blow to his team’s survival hopes.
According to the Press Association, when asked what his side needed to do to preserve their Premier League status, he told reporters: "Don't lose at Norwich, don't lose - they are seven points clear if we lose with five games to go. It still wouldn't be mathematically certain, but it starts heaping too much pressure on.
"We'll try to win, but make sure you don't lose because don't forget, we have still got a game in hand."
Sunday’s loss to Leicester stretched Sunderland’s winless run to six games, and was their second consecutive league outing without a goal.
And Allardyce admits a lack of cutting edge in the final third is proving to be a growing problem.
"The turning point was Jack Rodwell's miss, which is becoming one of our problems now, converting chances and here we are, lost the game, tried our very best playing against the top of the league.
"We weren't quite good enough and they punished us when they waited and pounced on the opportunity brilliantly."
Following their trip to Norwich, Sunderland face Arsenal, Stoke City, Chelsea, Everton and Watford.