The pair followed the Spaniard from Wigan to Goodison Park - Kone costing £6million with his team-mate arriving on a free transfer - but despite the Toffees' promising start to the campaign, neither has made an impact.
Injury has dogged Kone, with his latest knee complaint giving Martinez some cause for concern.
The striker, who turns 30 later this month, has managed just 79 minutes of Barclays Premier League action in five substitute appearances and has yet to score.
A hernia problem has prevented 31-year-old Paraguay defender Alcaraz making his debut so far.
Nevertheless, Martinez believes the pair will prove their worth to their new club - even if his comments regarding Kone suggest it may be some time before the Ivory Coast international regains full fitness.
"I felt pre-season was a challenging time for him, for different reasons but I don't think they were footballing reasons," said the manager.
"I think Arouna when he came on against Cardiff (on August 31) I felt he was ready and that game I saw him being himself.
"I thought when he came on against Hull (on October 19) he straightway had a massive effect on the game and he was ready.
"But we have had a little bit of a set-back and we are a bit worried with the knock he picked up against Hull.
"I am a bit concerned with his knee because he is someone who gets through anything and little niggles will never stop him from playing or being available.
"I have never seen him like this and we need to make sure we get to the root of the problem.
"We are going to assess him again and see because it is a little bit more serious than expected."
It seems likely Alcaraz could be named in the squad ahead of Kone, with Martinez claiming he will be available for the first match back after the next international break - which would be the Merseyside derby on November 23.
"I am convinced that Antolin is ready now," he said.
"Early on in the season he had two or three set-backs and we were a little bit concerned if we were going to find the solution to get him back to his best. Now he is 100 per cent on the road to being his best.
"The good thing about Antolin is he has tremendous experience at any level and has been in the British game for a long time.
"Once he is fully fit he will be ready to help the team."
Being fit and getting pitch time may be two different things for Alcaraz, as it seems highly unlikely he will displace established centre-back pair Phil Jagielka and Sylvain Distin.
"I don't see that as a problem," added Martinez.
"In football having injuries or a set-back from a fitness point of view can really stop you from being at your best.
"But when you are a top player and you have that real talent, once you are fully fit you are going to be a real positive to the squad and a real asset throughout the season.
"It is not a worry. When you have good players all you want is for them to compete, to push each other's levels in training, and that benefits the team.
"I would never have a concern about how many minutes a player is going to play, my concern is just to have everyone fully fit and ready to enjoy their football and the team will benefit from that.
"You cannot have enough experience and enough good players in your squad."