Lukaku, who scored four times against Bournemouth last weekend and is the Premier League's top scorer with 16 goals, is reportedly a target for former club Chelsea although his agent Mino Raiola said on Thursday he is close to agreeing a new contract with Everton.
"I didn't speak to the board about Romelu but if his agent has mentioned he likes to stay or is close to a new deal then that's perfect for Everton," Koeman told a news conference ahead of this weekend's away game at Middlesbrough.
"If players see a future at the club, if they see the team improving and getting stronger then that's always a plus for them to stay.
Lukaku signed a five-year deal with Everton in July 2014 after joining for a club record fee of 28 million pounds.
"Everyone has their own ambition, that's normal, and we like to keep the good players. At the moment he's the top scorer, playing at a high level, so clinical, and it would be difficult to find a replacement so I hope he stays," Koeman said.
Koeman told a Belgian newspaper in November that Lukaku may eventually need to look beyond the Merseyside club to fulfil his potential, suggesting that he could one day play for a team such as Barcelona.
Last season Lukaku became the first Everton player since Dave Hickson in 1954 to score in eight consecutive games.
He has continued that form this season, although Koeman believes there is much more to come from the 23-year-old.
"He's a world class finisher but in other details, other qualities, he can and must improve, he knows that," Koeman said.
"The team makes Romelu the top striker, obviously you need the striker to have that quality of finishing, but you can't do it by yourself."
Everton are seventh in the Premier League after a seven-match unbeaten run. A win against struggling Middlesbrough could see them close in on the top four. They are seven points behind fourth-placed Arsenal.
But Koeman warned against taking Middlesbrough lightly.
"They are a well-organised team, they will make it difficult, they only lost 1-0 away at Tottenham," he said.
"It's step by step. The first target now is to catch the team above us."
(Reporting by Martyn Herman; Editing by Hugh Lawson)
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