The Foxes sit bottom of the Barclays Premier League and are seven points from safety with just nine games left to play.
While the task facing Leicester boss Nigel Pearson grows more daunting every week, Allardyce feels he can take heart from the fact Sunderland were anchored to the bottom of the table with just four games remaining last year.
Leicester lost 4-3 at Tottenham before the international break and have held their own in most games this season - and Allardyce, who takes his West Ham side to the King Power Stadium on Saturday, thinks that is a good place to start a last-gasp battle against the drop.
"Nigel must be pulling his hair out to say 'why have those performances not given them more results?'," said Allardyce.
"I suppose it's the cruelty of the Premier League where teams haven't taken advantage of things, by not scoring chances, ultimately they have paid the price
That is this level.
"Somebody at this stage of season, generally, at that level, down at the lower end of the league, can win four out of six or three out of five and give themselves a lifeline
Last year was Sunderland.
"So that can happen with the performance that Leicester have shown
We have to make sure that it's not against us, we have to be on top of our game and deliver the performance we need to."
Sunderland were steered to safety by former boss Gus Poyet as he oversaw notable victories over away to both Chelsea and Manchester United and a draw at Manchester City.
Poyet has since been sacked as Sunderland could not build on that form and once again struggled this year - but Leicester have kept faith in Pearson and Allardyce feels that his former colleague at Newcastle has proved his worth.
"Nigel has handled things okay - he's his own man," he added.
"He has his own methods, he believes in what he's doing, he doesn't get too flustered by it
Occasionally, as we all get all under pressure, we may make the odd comment or odd mistake, but we are human beings.
"In general, he's done an unbelievable job as a manager
Later on in his career, he has been very successful.
"The job he has done at Leicester, the way they are playing, they are all disappointed where they are, they believe they can get more points on the board."
Source : PA
Source: PA