The Croatia international left for Hull this week and that leaves the Toffees light up front with the burden being placed solely on on-loan Chelsea striker Romelu Lukaku.
Out-of-favour Stoke forward Kenwyne Jones has been linked with a move to Goodison Park but Martinez would not comment on speculation even though he initially admitted: "We know who we want."
He later clarified that by saying he was referring to the type of player but regardless it seems unlikely there will be any imminent arrival.
"Obviously Niki has completed his move to Hull City and we wish him well," Martinez said.
"We are going to take our time but yes, my focus is to bring another attacking option that can help what we have got.
"We know exactly who we want and we are going to work hard to try and get the type of personality we want into the group.
"I am aware there is a list of at least 10 to 15 names but I will never speak about a player registered at another football club.
"In January it is not easy. You need to get many things together and what seems impossible one day, there can be an opportunity the day after.
"Normally it goes down to the final day of the window and I think it is going to be the case this time.
"I am quite happy to wait for the right person. I don't want to bring in a player just for the sake of it.
"When I say 'We know who we want' I mean the profile of the player, what type of player we need, because we are very strong in many areas.
"It is very clear the type of player we need.
"The next few months is a demanding period with two competitions to face and we don't want to go through games hoping for things to happen, and for that you need extra cover in certain positions."
The departure of Jelavic has been offset by the arrival of Aiden McGeady from Spartak Moscow and he will go straight into the squad despite lacking match-fitness after joining the club's warm-weather break in Tenerife this week.
"I am delighted with Aiden McGeady, someone who fits in our football club so well," said Martinez.
"I never expected him to be working with us in January if I'm honest.
"For Aiden to be able to spend those 24 hours a day with Everton, getting to know the squad and how we work was very important.
"He hasn't trained in the last few weeks and hasn't done any group work in the last six weeks so he needed the extra work but it has been perfect.
"It has been like a mini pre-season for him and now he is ready to go straight into the squad.
"He is not a player you would expect to have 90 minutes but I do feel he is ready to be part of the squad."