There was a concern the hamstring injury which saw the the Ivory Coast international withdrawn during the warm-up at Chelsea on February 22 could see his stay at Goodison Park over after just an hour's action if the problem required surgery.
An operation is not necessary and, although the 6ft 8ins forward will still be sidelined for a considerable time, Martinez hopes to have the player available for the run-in.
"With Lacina, we've seen the specialists and we're now ready to proceed with what's going to be the best treatment for Lacina, and it's not going to be surgery, which is fantastic news," Martinez told evertontv.
"It's fantastic news that he doesn't need to go down that route, as that means he's going to be available for the end of the season.
"We're expecting between six and eight weeks to be the period he needs to be recovering, but we're all very positive."
Martinez has not given up hope of having captain Phil Jagielka back to face Arsenal in Saturday's FA Cup quarter-final.
The England centre-back was ruled out of Wednesday's friendly against Denmark because of a hamstring injury which caused him to miss last weekend's win over West Ham.
"With Phil, you can never rule him out. From a medical point of view, he should be missing the game against Arsenal - I was probably looking more at the Cardiff game (on March 15)," added the Toffees boss.
"But knowing him and how he's reacted to this similar injury that he's had in the past, he could heal a lot quicker.
"I'm not going to rule him out and we're going to give him as much time as he needs to make a final decision.
"In a way it's good that he's behind-the-scenes here at Finch Farm and he hasn't got the game with the national team.
"That could give him a small percentage (chance) to be fit for the weekend.
We'll see how this week unfolds and make a final decision closer to the game."