Allardyce's men are seven points clear of the Premier League drop zone with five games to play following a run that has seen them lose just four of their last 11 matches.
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"The success of the teams below us over the weekend and the results that were surprisingly pulled off mean that we still look down because we're not mathematically safe," he told reporters.
Arsenal - who won Saturday's FA Cup semi-final with Wigan on penalties - head into the game with a number of injury concerns and without a victory in four Premier League matches.
However, despite their problems, Allardyce knows his side will still face a difficult task to take all three points, especially with captain Kevin Nolan doubtful due to a hamstring problem.
"[Arsenal] keep a squad of 25 international players and they're playing at the Emirates, so whatever difficulties they have it's still a hugely tough fixture for ourselves," he added.
"Whatever team Arsene [Wenger] puts out, we still have to play at our best to get a result, particularly on their own patch.
"Kevin Nolan is our problem at the moment. He came off with a hamstring against Liverpool, obviously that's one that we have to be very careful about.
"It's unusual for Kevin to sustain a muscle injury, I think it's the first of his career, but we'll be very careful about how we handle him.
"Kevin is going to desperately want to play, so we'll probably wait until tomorrow to see how he is."