The Black Cats are desperate to turn draws into wins but Claudio Ranieri's men will be tough opponents.
The Foxes, who are seven points clear at the top, travel to the Stadium of Light on Sunday with the Wearsiders' survival hopes fading by the week.
Four points now separate Sunderland from 17th-place Norwich City, who host them next weekend.
A draw against Leicester, who've won five of their last six matches, would not be the worst result if Norwich fail to beat Crystal Palace on Saturday but Allardyce says he'll go for all three if his game is heading towards a stalemate.
"On the form that they’re in and what they’ve achieved over the last five games, you’d have to say yes (we would take a point)," the 61-year-old told the Alan Brazil Sports Breakfast. "But we do have to try, and will at some stage, to go all out to try and win it.
"If we’re in a position where we’re drawing with 10 or 15, 20 minutes to go, we will have a go because we’ve got to. We’ve got to try and secure those three points and make sure we catch up on the rest."
That could be easier said than done, however, with Sunderland failing to win any of their last five matches - though four of those were drawn.
Leicester, who are looking for their fifth successive clean sheet when they travel to the North East, need 12 points from their remaining six fixtures to secure their first ever top-flight title.
Sunderland were beaten 4-2 by the Foxes on the opening day of the campaign, though both teams have tightened up since then.