The Toffees are struggling in both the Premier League and Europa League, with just two wins from their last 10 games in all competitions.
Koeman spent over £150m on new players in the summer, although the club did recoup £75m by selling star striker Romelu Lukaku to Manchester United.
Despite starting the season with high expectations, the Toffees only sit 16th in the top-flight standings.
The Dutchman has insisted that he will remain "calm" as he looks to turn around the club's fortunes.
"Everyone is entitled to their opinion but as a coach, I don't run away from criticism,” Koeman said in his column for De Telegraaf.
"We know that we our not playing at our normal level and the results should be better.
“When I made the move from Southampton to a club with such ambitions, I knew I could be in a difficult situation if the results were not there.
"I am calm and we will work with my staff to get those results.
"I am exactly the same coach as when it was going well."
Bookmakers have shortened the odds on Koeman being the next manager to be sacked in the Premier League.
The 54-year-old had a win ratio of over 60 per cent during his time in charge of Ajax and PSV.
But since taking charge of Everton in June 2016, Koeman has led his side to just 24 wins from 55 games, which equals a win percentage of under 44.
Record signing Gylfi Sigurdsson who joined from Swansea for £44m has disappointed so far, registering just one goal an one assist in his nine appearances.
Everton travel to Brighton next weekend for a Sunday lunchtime kick-off (13:30).
