After the first seven games, the Toffees find themselves with just seven points on the board and sit two points above the relegation zone, while they have taken just one point from their first two games in the Europa League.
Koeman, who oversaw spending of £133m in the summer transfer window, has found his position under scrutiny as a result but the Dutchman says that it is part and parcel of his job.
"There's no extra pressure for me," he told reporters this afternoon ahead of his side's trip to Brighton & Hove Albion this weekend. "I think the pressure is always there in football, because you like to win and you put yourself under pressure to win the game.
"Everything which is mentioned from outside is out of hands; I can't stop it and I can't do anything about that. Every Premier League game is a tough one regardless of the name of the opponent."
"I'm not surprised that they have put my name in because they did the same some years ago, so it's not a surprise," he added. "But I'm Everton manager and that's the most important thing and nothing else.
"The future is not realistic now. We have other business to do at Everton. We need to win and we need to change our situation and our position in the table and that starts on Sunday."
After the trip to the south coast, the Toffees welcome Lyon to Goodison Park on Thursday evening.