The lofty striker was withdrawn minutes before Saturday's defeat at Chelsea with a hamstring injury sustained in the warm-up and may not feature for the Toffees again this term
The striker arrived at Goodison Park on loan from Monaco in the January transfer window and scored on his debut in a 3-1 FA Cup win over Swansea City earlier this month.
However, the Ivorian suffered a hamstring injury in the warm-up before Saturday's 1-0 defeat against Chelsea, and both Monaco and Everton have agreed that Traore should visit experts in both countries as they seek a diagnosis.
If he requires surgery, his loan spell may be terminated, although if the injury is less serious he will remain at the Premier League club for the remainder of the campaign.
"We have been working closely with Monaco to ensure we get the best possible treatment on the injury," Everton boss Roberto Martinez told the club's official website.
"He will now see two specialists next week - one in London and one in Paris. Then we will make the final decision with the parent club.
"If Lacina will be fit by the end of the season, then we will see him in an Everton shirt again, and that is what we are aiming for."
Martinez's men host West Ham at 15:00CET on Saturday as they look for a fifth successive Premier League home win to keep their top-four hopes alive.