The 30-year-old defender, who has won six caps for his country and last played for Red Star Belgrade, has been training at Hillsborough for much of the season and has featured in a number of their pre-season friendlies.
He did not feature in Saturday's 1-1 draw at Barnsley, but boss Stuart Gray confirmed that he is considering the offer.
"We are offering Dejan a short-term contract and hopefully he will be putting pen to paper for us," Gray said. "So suddenly, I have got some competition.
"Nothing has been finalised, he is mulling it over at the moment.
"He's been with the group a good few weeks, he's fitted in well and he's done well."
Gray is hoping to bring in more players before their Championship season kicks-off at Brighton on Saturday, with a striker on top of his list of priorities.
And the 54-year-old has a target in mind.
"It's someone we have got in mind and we are probably competing with other clubs to get him in," he added. "But at the moment his parent club don't want to let him go, but we will get a final decision soon which I am sure will be a positive one.
"I would love to say it is 100 per cent certain it is going to happen but you can never say 100 per cent.
"We need a forward in, touch wood it will happen before next Saturday.
"I will only bring players in that will improve the squad. I am always looking to bring players in, you need that competition."
Gray confirmed the Owls will not be offering a deal to trialist Roy Contout.