"I didn't speak to the board about Romelu. If his agent says he likes to stay and is close to signing a deal then that is perfect for Everton," Koeman said.
"Normally players, if they really see a future in a club, they see the team improving, the team is getting stronger every day, then that is always a plus for players to stay.
"We know everybody has his own ambition, that is normal, that's a human being, and we like to keep the good players and he is one of them."
Lukaku has scored 16 goals in 23 Premier League appearances this season - form that has helped the Toffees climb to seventh in the division.
Koeman admits it will be difficult for Everton to replace the former Chelsea striker if he decides to leave the club in the summer.
"At the moment, he is top scorer in the league and playing at a high level. He is so clinical and it would be difficult to find a replacement for him. I hope he will stay," Koeman said.
"First of all, it starts by yourself, it is to improve. He is really a world class finisher but in other detail, in other qualities, he can and must improve and he knows that's the most important thing.
"If the club and team is improving and showing ambition then it is always a good way to stay for players and that is what we need to show him that he is convinced that the short future is at Everton."
Lukaku will be hoping to add to his goal tally when Everton travel to face Middlesbrough on Saturday afternoon.
Aitor Karanka's side have not won in the Premier League since their 3-0 victory over Swansea on December 17.