The Ivory Coast international reported a problem during the warm-up ahead of Saturday's defeat at Chelsea and was immediately withdrawn from the starting line-up.
"Obviously with this sort of injury it is normally weeks rather than months," Martinez told City Talk 105.9.
"The type of player Lacina is, a sprinter, means these sort of injuries need to be well recovered."
The 6ft 8in forward arrived from Monaco, having not played since November because of the winter break in the Russian season while at previous side Anzhi Makhachkala, on January 24 with an existing hamstring injury to his other leg.
That delayed his involvement until he scored on his debut in the FA Cup victory over Swansea on February 15, which remains his only appearance.
He now looks like missing possibly the majority of March, so the return to fitness of Everton's other on-loan striker Romelu Lukaku is timely.
The Belgium international injured an ankle in the Merseyside derby defeat at Anfield a month ago but would have been fit to play last weekend had the match not been against parent club Chelsea.