The Toffees are currently sixth in the league table with 54 points from 29 matches, eight points behind Arsenal, who are fourth and occupy the final Champions League place.
However, the Merseyside outfit have a game in hand over the Gunners and will be looking to close the gap on Tuesday when they take on Newcastle United at St James' Park, while Arsenal host Swansea City.
McCarthy believes that Arsenal should be wary of Everton and insists that he and his team-mates will not give up on a top-four finish until it is mathematically impossible.
"They (Arsenal) will know they cannot let their standards slip," said the 23-year-old.
"They will be looking over their shoulder now. Obviously they will be a bit worried.
"To be honest we never wrote ourselves off [in the Champions League race].
"We will keep playing until it is mathematically not possible. It's still open and we will give it our best shot.
"We are still confident we can achieve something big here this season.
"We need to get a few results from now until the end of the season and hope that results go for us elsewhere but we will take it from there."