The Pro Bowl outside linebacker has been dealing with a back injury that predates the start of training camp last month.
Smith had been on the non-football injury list until last Monday, when he was activated and practiced on a limited basis. But by the end of last week, he was back on the sideline.
On Sunday, coach Matt LaFleur said Smith "could potentially" miss the Week 1 game at the New Orleans Saints on Sept. 12.
"I'm not quite sure when we'll get 'Z' back," LaFleur said Sunday. "He's injured right now. I know he wants to be out there, but we're going to be smart with him. He's obviously a big-time player for us and, yeah, it definitely is a concern."
Smith, who has made two Pro Bowls in as many seasons with the Packers, is one of only three players in the NFL with 12-plus sacks in each of the past two seasons, along with Aaron Donald and T.J. Watt.
Smith has not missed a game since Week 14 of the 2017 season.
"Obviously he's working through something right now and I'm hopeful that that's not the case, as far as missing Game 1, but he's dealing with his little day-to-day [injury]," Packers general manager Brian Gutekunst said Sunday.
"We got him out there to ... see if he could get a couple good days [of practice last week], and then it just flared up on him again."
Gutekunst insisted that Smith's absence from practice is not related to his contract.
The Packers restructured Smith's contract in March, when they converted $14.26 million of his existing base salary and roster bonus into a signing bonus, but that was purely for salary-cap reasons. No new money was added to his deal, and Smith is believed to be seeking a contract extension.
Gutekunst also said Sunday that they "moved some money around" in Smith's deal right before training camp started, but it's not clear what -- or how much -- was moved.
Smith is entering the third year of a four-year, $66 million contract he signed with the Packers in free agency before the 2019 season.
Fellow outside linebacker Rashan Gary (a first-round pick in 2019) also has been in and out of practice with a groin injury, leaving the Packers short-handed at the position during training camp practices.
LaFleur also said Sunday that quarterback Jordan Love could return to practice after missing all of last week with a right shoulder injury. The Packers would like to get Love another preseason start. He played the first half of the preseason opener but did not play against the New York Jets on Saturday.
The Packers' plan is for Love to do some individual drills in practice on Monday before they put him back in 11-on-11 drills. LaFleur said Love threw every day last week even though he did not practice.
"It's just trying to get the volume up, trying to get the velocity up [and the] distance just to see how he feels," LaFleur said.