Sunday's 2-0 loss at Arsenal left Everton six points clear of the bottom three heading into the midweek trip to Stoke City.
Roberto Martinez's side have won just once in the last 11 top-flight games and, although Barry remains confident, he is not getting too comfortable just yet.
The midfielder said: "In terms of performance and belief in the dressing room we don't feel like we have any sort of danger or talk of relegation.
"But we are aware that if you don't start putting points on the board then you can get dragged in there.
"The quicker we do that the better it will be for the club, and we can concentrate on winning the Europa League and finishing as high up the Premier League table as possible."
Everton are through to the last 16 of the UEFA Europa League, where they will face Dynamo Kiev over two legs later this month.
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