Defenders Phil Jagielka and Michael Keane will be given every chance to be fit for the visit of Burnley on Sunday.
The pair both missed the win over Bournemouth last weekend and Thursday's Europa League draw with Apollon Limassol, with the Toffees skipper also sitting out against Sunderland in the Carabao Cup.
Jagielka (hamstring) and Keane (foot) have both been working hard in order to be fit to face the Clarets, but Koeman revealed a decision will be made on the defensive duo on Saturday.
“I hope tomorrow I will have some positive news,” said the Blues boss. “We have to make a decision about Michael and Phil over whether they are available for Sunday. They are doubtful.
“First of all, it is important to have everybody available, because those we have out are quality, experienced players. That is what you need in a difficult situation.
“We will put out the best XI possible, with the most fresh players, in order to try to win the game.”
Koeman also revealed Wayne Rooney has been passed fit to face Sean Dyche’s men.
The forward injured a hand in the 2-2 draw with Cypriot side Limassol on Thursday night but tests have disclosed no damage.
However, James McCarthy continues to miss out on selection, with the Blues boss suggesting the midfielder was set for a spell with David Unsworth’s Under-23s in order to build up match fitness.
“Wayne had a scan of his hand this morning,” added Koeman. “There is no fracture and he is okay.
“James is not available. He did the first part of the session today but not the rest of it. He needs game time first with the Under-23s. That’s how players need to come back from a long time out injured.”