The Black Cats currently sit second bottom on the Premier League table, five points from safety, but Allardyce feels there is enough quality at the Stadium of Light to climb out of the mire.
However, Big Sam is well aware that his players will need to show plenty of heart and desire to actually achieve the feat rather than him simply saying it is possible.
"I think we can avoid the drop, but we are going to have to fight to the end now, based on the amount of points we need to catch up, and the games we have left," said Allardyce ahead of Saturday's home clash with Manchester United.
"It's hard to say how many points will get you there but we could say probably 40, this time round, will be safe. It's been a long time since anyone got relegated with 40 points, but 38 might be enough.
"If we can put up the same show we put up against Manchester City [a spirited 1-0 defeat], we're going to give them a really good game, but we've got to get the result along with it.
"It's the most important thing now, not how we get it.
"Providing chances for Jermain Defoe is critical for us. But our first major problem is that we haven't kept a clean sheet in nine matches.
"We need to start putting a zero against us in games and if we start doing that we'll win more matches. If we win more matches, we'll get out of trouble."