Sunderland have one point from their first three games and Moyes warned after last month’s 2-1 derby loss at home to Middlesbrough that the Black Cats are likely to be in another relegation battle.
Koeman said: “He showed he was a great manager for Everton in all the years he was in charge. Sunderland is a different club. The last few years they have been struggling — and there is what he said himself about relegation.”
Koeman — who has seen Everton take seven points from three games since arriving from Southampton — said: “Sunderland are also a club that have signed new players, and the team will be different than it was two weeks ago. It is a difficult one - we need to be prepared for that.”
Having achieved European football in his first two seasons in English football with Southampton, Koeman targets the same for his new club, who were 11th the last two seasons.
“Everton needs to be a team who fight for Europe and I think we have the squad available to do that,” he said. “The start is OK, but now we need to continue.”