Schwarzer made the comments following Everton’s defeat to Watford away from home in the Premier League at the weekend.
Ronald Koeman’s side went down 3-2 to the Hornets at Vicarage Road on Saturday afternoon.
Schwarzer believes that Everton should be worried as they have a tough run of games coming up.
“Looking at the run of games they have to come, Ronald Koeman and their fans will be extremely worried,” the former Australia international told BBC Sport (14:25).
Everton are ninth in the Premier League table at the moment with 20 points from 15 matches.
The Merseyside outfit are 10 points behind Manchester City, who are currently fourth in the standings and occupy the final Champions League spot.
The Toffees will return to action on Tuesday evening when they take on Arsenal at Goodison Park in the Premier League.
The Gunners are in good form at the moment and will head into the match as favourites.
Then on next Monday, Everton will host Liverpool at Goodison Park in the Merseyside derby.
The Reds will certainly fancy their chances of picking up all three points from the Premier League game.
Koeman’s side will then face last season’s Premier League champions Leicester City before locking horns with relegation candidates Hull City away from home at the KCOM Stadium.