The 61-year-old, who has built his reputation on stabilising and rebuilding struggling clubs, saw the Black Cats slip to a third defeat of his four-game reign to date when they went down 1-0 to Southampton at the Stadium of Light on Saturday.
They remain entrenched in the bottom three with just six points from their opening 12 games, and Allardyce does not accept that he has plenty of time to turn things around.
He said: "No, I haven't, no
We are relegation favourites because we are in the bottom three and getting out of the bottom three is a long haul now.
"When you have got fewer points than games, as many fewer points than games as we have got, then we have got a mammoth task just to get out of the trouble we are in at the moment.
"There's no point in trying to bull**** anybody: it's going to be a long haul to get where we need to get.
"What I want to try to do is not have one of those end-of-the-season desperations that this club has become used to over the last four years where if you don't win four out of the last six, you are going to get relegated.
"That's the last thing that these players should have on their minds
Now is the time to start winning and drawing, not the last six games of the season, right now starting next game."
That next opportunity will come at Crystal Palace after the international break when Allardyce will once again attempt to strike a balance between defensive resilience and attacking flair.
Having seen his side ship six at Everton last weekend, the Sunderland boss understandably adopted a more cautious approach against the Saints, and while that paid dividends at the back, it meant they lacked the cutting edge to threaten the visitors.
However, they might have emerged with a point had midfielder Yann M'Vila not needlessly felled full-back Ryan Bertrand inside the box to hand Dusan Tadic the chance to win the game from the penalty spot 21 minutes from time.
Allardyce said: "It's just a shame he didn't control his emotions because he [Bertrand] was going nowhere, he wasn't going to do anything, he wasn't going to harm us.
"Just take him out of the box and just shuffle him away, and unfortunately he made the decision not to."
Southampton, who cemented themselves in seventh place in the table as a result of their win on Wearside, are unbeaten away from home in the league this season, and manager Ronald Koeman insists they will continue to play the same way wherever they are in action.
He said: "It's important, important for the confidence, important for the way we like to play
We don't play differently at home than we like to do away
I think that's the best way to be constantly.
"Okay, we had the chance to have a lot of ball possession here from the start of the game, and then it's to be patient.
"You can compare maybe this game to the West Brom game away, and it's not easy to create
But I think the level in front was higher and that makes the point."
Source : PA
Source: PA