Lukaku scored twice as Everton won 4-2 in the Premier League at Goodison Park on Sunday, increasing his top-flight tally to 23, four clear of Harry Kane, his closest rival for the Golden Boot.
The striker is under contract until 2019 but has reportedly turned down a lucrative extension with the club and faces an uncertain future amid speculation linking him with former side Chelsea.
Koeman hopes to persuade star duo Lukaku and Ross Barkley – whose own deal has just over a year to run – to remain with Everton, but could not make any promises to the supporters.
"We will try [to keep them] but the final decision is always by the player," he said to Sky Sports.
"Of course the situation with Ross is different because he has one more year of contract and Rom has two more years of contract, so it is a different situation.
"We try to do everything to gather a really strong team to get really competitive with the big six teams in the Premier League. What we have shown until now in our first season is the first step and that will continue for the next season.
"They will have a picture of that belief and the direction we will go in - that is very clear to everybody."
7 - Romelu Lukaku has now scored in seven successive Premier League appearances at Goodison Park (11 goals). Phenomenal. pic.twitter.com/nB02zgoxZ8— OptaJoe (@OptaJoe) April 9, 2017
Koeman added to BBC Sport: "Romelu Lukaku is one of the best strikers in the world.
"Everybody knows he is a key player for Everton and we will try to do everything to keep Rom here at Everton but the final decision is always with the player.
"Of course he has two more years' contract but everybody in football has his own ambition in football. We will try to do everything to keep him.
"I cannot change anything in him about how he finishes opportunities and scores goals because in my opinion he is the best finisher. He is improving other aspects that the team needs and he is going in a good direction."
26 - Everton have already scored more league goals at Goodison Park in 2017 (26 in 7 games) than in 2016 (25 in 18 games). Entertainment. pic.twitter.com/6YpEycIJl6— OptaJoe (@OptaJoe) April 9, 2017
The Everton manager was impressed by the displays of both Lukaku and Barkley and did not think they had anything to prove despite being criticised heavily after last week's Merseyside derby defeat against Liverpool at Anfield.
"It is not about them needing to respond," he said. "They need to play at the best level they can show and they normally do.
"I have defended the boys against criticism and they showed they are fantastic football players.
"The early cross from Ross and the finish from Romelu [for Everton's second goal] was fantastic. They were two players on a high level but I think the whole team played on a high football level and that is what we like to see at home.
"I'm really happy with the three points and how we played. We played some fantastic football in the first 45 minutes.
"Then, after it went 4-2, we played it really clever and controlled the game. It was a really good team performance."