The Belgium international was believed to be on the verge of a return to former club Chelsea, but decided to stay put in Merseyside.
In an interview conducted with Thierry Henry on Sky Sports News, Lukaku said: "I had a moment of reflection - for a week in pre-season - to see how the manager works and see what he sees in me as a player.
"I'm young and there are aspects of my game I need to improve. He worked with me on it from day one and I was convinced he was a man that can have a big influence on my future.
"I know that if I don't do what he wants me to on the pitch, I can go on the bench straight away. He keeps me very sharp and focused in the training sessions and throughout the games which is going to help me."
Koeman, formerly at the helm of Southampton, took over at Everton this summer after the sacking of Roberto Martinez.