The Maccabi Haifa stopper was on Wearside last week having trials for the Black Cats, hoping to tie up a permanent move.
But he has returned home with no deal agreed and is unlikely to be coming to the Stadium of Light in the near future.
Manager Mick McCarthy has not ruled out moving for the Israeli international in the January transfer window but admits that no firm plans are being made.
He said: "Davidovich has returned home. We plan to keep tabs on him – he's due to play a game for his club on New Year's Day - but we have no plans to take things further at this stage.
"I wouldn't want to rule him altogether though because he is one we're looking at. As it stands, I'm not thinking about taking it further with Davidovich but that's because I'm still looking at a number of options and there's still a lot of uncertainty."
McCarthy told the Echo that Davidovich is just one of a host of players who have been on trial at the Academy in recent weeks.
He said: "We've looked at quite a few players, Davidovich among them, but it is difficult to guage their true quality with the way the rules stand at the moment.
"We had a lad called Ricardo over from Venezuela who looked useful but it is so hard to tell how players will play in match conditions and we're not allowed to play them at the moment in a reserves game.
"It's a real beef I've got that we can't do that. It is complete and utter nonsense and just about the biggest bit of rubbish I've come across in football. Why we can't play them in reserves games? You could set a limit on it – one, two or three trialists; you could set a limit on the number of games, even if it's just one or two.
"But let's see them play and let the fans see them play too, rather than just seeing them in training – even a practice match in training isn't really a competitive game. I would like nothing better than to be able to put them in a match and see how they perform but in training it is always hard to be sure.
"They come here for training and obviously they don't want to go steaming into challenges because my players could get injured and I would get the hump with them and that's not what I'm after.
"I wouldn't mind if they steam into challenges against other club's players, in a reserve match for example, I wouldn't mind that at all and I'd also be able to get a better idea of what they're all about. But as the rules stand I'm not able to do that and I think that's rubbish."
McCarthy has little room for manoeuvre in the January transfer window with club chairman Bob Murray already signalling that little money is available for a move into the market.
But the Sunderland boss might have some scope in the transfer market and is hoping to improve his squad without weakening it by selling players.