The striker, who arrived on loan from Monaco in January, suffered a hamstring injury during the warm-up at Goodison Park prior to Saturday's match against Chelsea, which the Toffees lost 1-0.
Having been withdrawn from Martinez's line-up and replaced by Steven Naismith, Traore underwent scans and is now facing a lengthy lay-off, with the boss confirming that he will be out for "weeks rather than months" and conceding that the news comes as a significant blow to his seventh-placed side as they seek to seal European qualification.
|11/2||Everton are 11/2 with Bet365 to fail to score against West Ham|
"It's a really disappointing outcome. He was ready and had been working really hard since he arrived at the club."
Traore has been hampered by fitness problems since moving to England and has yet to represent the club in the Premier League.
Martinez continued: "Starting the game at Chelsea was the next target after Swansea [against whom he scored in the FA Cup] and for him to pull his hamstring in the last action of the warm-up was a really disappointing moment.
"We are a little bit concerned. We are working with his parent club and making sure we get the right diagnosis.
"Lacina was really down after and we need to support him. We will find out over the next 48 hours how we will treat the injury. In those sort of injuries, similar to the injury Gerard Deulofeu picked up before Christmas, the soft tissue is an area which needs to be well healed."
Everton can at least take heart from the imminent return of another on-loan forward, Romelu Lukaku, who has been absent for a month with an ankle injury.
"Romelu has been working really well and he is looking fresh and ready," added Martinez. "We are all very excited to see him back enjoying his football with really top energy levels."