Rangers chief executive Martin Bain failed to seal a deal for Rapid Vienna's Croatia international striker Jelavic after flying to Austria this week, but on Friday Smith revealed that talks were continuing.
"Negotiations are ongoing," Smith said. "They are keen for him to play in their domestic game at the weekend.
"We will hopefully find out one way or another at the start of next week."
The 24-year-old scored 21 goals last season, including three against Celtic in the Europa League.
Smith added: "Negotiations with both the club and the player, and his agents, are ongoing.
"He's a player we have been watching for a while. As normal, the club initially weren't too keen to sell him, but then they indicated they would be happy to listen to what we had to say.
"He's a player I think would do a really good job for us."
Smith does not expect Rapid to hold out until after their Europa League play-offs later this month.
"No, there's not that fear," Smith said. "I think we have to have a resolution at some stage next week."
Meanwhile, midfielder John Fleck faces six weeks on the sidelines after suffering a hamstring injury in Wednesday's 2-1 win over Clyde.
Rangers had expected the timeframe to be two to three weeks but Smith said: "Unfortunately he will miss maybe six weeks, up to that anyway. Hopefully we will get him back before then.
"It's disappointing because he has been playing well in pre-season."
Smith, who will be without Andy Webster for tomorrow's friendly against Newcastle after the defender suffered a groin strain, also gave an update on Honduran trialist Georgie Welcome.
"He has only been in a couple of days of full training," the Rangers boss said.
"Unfortunately we can't play him in any games.
"Hopefully we can get a closed-door game in the early part of next week and look at him then."