Belgium international Lukaku, who struck 18 Premier League goals last season, has continued his impressive form under Koeman, netting six of Everton's 12 league goals.
Koeman's side, who are sixth in the standings and have conceded just six goals after the opening eight league games, boast the second best defensive record in the top flight after leaders Manchester City.
"Of course, you'd like many players to score more goals. The players around Lukaku need to show more productivity. There is no pressure on Lukaku. It's one of his best qualities," the Dutchman told a news conference on Thursday.
"On the other hand we haven't conceded more than one every game but we need to create more and we need to score more goals."
The 53-year-old Koeman added that midfielder Ross Barkley's omission from last weekend's clash with City could be a wake-up call for him.
However, Koeman, who last month told Barkley to improve "in all areas", said the 22-year-old did not play at the Etihad Stadium for tactical reasons.
"They are quality players. He is young but he is not so young because he has been playing four years in the first team. It means you are not a talent anymore and need to show what you can do," he said.
"I changed him and I did not start with him but it can be different this weekend. If I am not totally happy or it is tactical then he does not start. Maybe last weekend was a wake-up call for him."
Koeman confirmed that left back Leighton Baines will miss Saturday's trip to Burnley after suffering a setback with his hamstring injury, while winger Aaron Lennon is available for selection after missing out against City.
Burnley manager Sean Dyche said injured midfielder Steven Defour will miss Saturday's game.
The 28-year-old, who has racked up three assists and scored once in seven league games this season, sustained a hamstring injury in 14th-place Burnley's defeat by Southampton on Sunday.
"Steven has got a pretty straight-forward hamstring injury. It's straight-forward as regards the scan. When players come off that quickly you are also a bit wary that it could be worse than it is," Dyche said.
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