The Premier League's most lethal marksman Lukaku continued his Goodison Park scoring spree by netting for the eighth straight home game with a brace in a 4-2 victory over Leicester, who failed to win for the first time since Craig Shakespeare replaced Claudio Ranieri in February.
Lukaku leads the way in the top flight with 23 goals, one more than Middlesbrough's entire haul, and the Belgian has been publicly vocal about his desire to win silverware and play in the Champions League.
As for homegrown talent Barkley, his deal runs out next summer and Koeman admits the scenario with the 23-year-old Liverpudlian is straightforward.
"We offer him a new contract, and then (there are) two possibilities," he said when asked what he could do to keep Barkley.
"One, he signs that contract, if he doesn't sign that contract then we need to sell the player. It's simple, it's not so difficult in my opinion."
Any Barkley admirers would have been encouraged by his performance on Sunday, though he was not alone in impressing in a blue shirt.
Tom Davies, who has recently penned a new deal, scored inside 30 seconds and though efforts from Islam Slimani and Marc Albrighton had Leicester ahead by the 10-minute mark, Lukaku's brace sandwiched a Phil Jagielka header to make it seven home league wins in a row for Europa League-chasing Everton.
Much of that owes to the form of Lukaku, who has scored 12 times in that run, and the reborn Barkley.
"We try to keep the best players," Koeman added.
"We spoke a lot about Ross and Rom because they are really important. Most of the time the quality of the players can be the difference between Everton and the opponent, and they played really well.
Read the full article on eurosport.co.uk: Koeman warns Barkley: Sign a contract or we'll sell you