The Wearsiders are currently in 18th place and four points adrift of Saturday's opponents Norwich City following a 2-0 defeat at home to title-chasing Leicester City last week.
Defeat at Carrow Road is likely to cast significant doubts on Sunderland's Premier League status and Allardyce, who achieved promotion from the Championship with West Ham United in 2012, admits relegation from the top flight would be painful.
According to TalkSport, he told reporters: "It would hurt immensely if the club got relegated, for the club's sake first, not mine. I've had a fantastic career.
"I would be disappointed if I had that one relegation against my managerial career from the Premier League. It would hurt, yes, but it's more important for the club than it is for me. As a football club, it's more important for the players than it is for me in terms of my career."
A first relegation from the Premier League since 2006 would inevitably lead to a number of departures from the Stadium of Light, but Allardyce has warned his players that the grass may not always be greener, adding that many of them would be better suited to staying in the north east.
"My career will take whatever way and whatever form it needs to take; the players, most of them will never get a club better than this, in my opinion, with its facilities and its supporters, playing in the top league in the world, and they really need to realise that.
"I can't see you lads getting a better club than this one, so let's stay in the Premier League and let's make the club even better. That's their big challenge in the next six games."
According to the Daily Mail, the calls to stay at the club from the former Newcastle boss have been reciprocated by goalkeeper Vito Mannone, who claims Allardyce has to stay even if they do slip out of the Premier League.