The 30-year-old has not played since sustaining a knee injury in October which has restricted his Toffees career to five Barclays Premier League substitute appearances, one Capital One Cup start and no goals after joining from Wigan in July last year.
Kone is still several weeks away from full fitness and although he is training with the rest of the first team, he has not yet managed to be involved in a full week's schedule.
While Martinez has plenty of players who can operate just off the forward line, Kone is his only real back up to first-choice centre forward Romelu Lukaku after he made last season's loan spell a permanent move this summer in a £28million deal from Chelsea.
The manager insists he is not desperate to bring in anyone else and is prepared to give Kone a chance to prove his worth - even if he will not be available for selection until the transfer window has closed.
Asked whether that was a gamble which could leave himself short, Martinez said: "That's like saying 'What if you get the two goalkeepers injured?'
Those situations you have to look into when it happens.
"It is important the players we have at the club have an opportunity to show what they can do.
"I don't want to bring in players just for the sake of it just because the position looks light.
"With Arouna it is difficult to say when he is going to be fit but I have an incredible confidence and belief in his ability as a footballer.
"I want to make sure Arouna Kone is in a good position so we can make a good decision from now until January really.
"A lot will be down to how Arouna adapts to being fully fit
It is a big area we are going to keep an eye on but very soon I think we will be very strong in that position.
"We are desperate to see him fully fit
He hasn't been able to train every day and that is where he is at this stage of the recovery.
"It is a question of a few weeks until he can be out on the pitch but there is no truth whatsoever he is for sale."
Martinez has added Christian Atsu to his attacking options after the Chelsea winger joined on loan but with Lukaku and Gareth Barry both making their loans permanent and £4million midfielder Muhamed Besic signing from Ferencvaros there are unlikely to be many more arrivals.
Those signings, plus the return from injury of midfielder Darron Gibson, gives Martinez more depth as Everton face domestic and European campaigns.
"I am very happy, today, at the start of the season, with the business we have done," said the Spaniard ahead of his side's trip to Leicester for their Premier League opener.
"We have brought the right characters into the club
Now we are in a position where I am desperate to see how the team reacts to the first game of the season
"If there is anything (in the transfer market) which will help the team until the next window opens we will always look into that but in terms of the squad we have I am delighted with what we achieved and the continuity we got.
"In terms of being desperate to bring someone in the answer is 'no' but we are going to use the period from now until the window closes to see how strong the squad is.
"If there is anything we need to help the players we will do that."
Atsu admits he seized the chance to join Everton as he does not know when he will get his chance at Chelsea.
However, the 22-year-old, who joined Chelsea on a five-year contract from Porto last September and was immediately farmed out to Dutch side Vitesse, is not yet thinking about long-term plans
"I can't tell you the future but I don't know how long it will take for me to get back to Chelsea," said the Ghana international.
"Looking at the players Chelsea has my playing time would not be much and for me I have to play football as a young player.
"Right now I am really focused on Everton and at the end of the season we will see what will happen.
"This is my first season in the Premier League and it is a really high level for me and a big opportunity to prove myself."
Source : PA
Source: PA