The 29-year-old has not resumed training with the first-team but is working on his own.
He won’t be ready for the Scottish Cup tie with Cove Rangers this weekend but assistant boss Gary McAllister confirmed he is getting “closer”.
There has also been progress for Nikola Katic who has been out since suffering a serious knee injury in pre-season.
Meanwhile, Ryan Jack has been to a specialist due to injury issues. The midfielder had to pull out of the recent Scotland squad and was sorely missed in draws against Austria and Israel.
Ibrox assistant boss Gary McAllister, speaking ahead of Rangers’ Scottish Cup clash with Cove Rangers on Sunday, said: "The positive on Tav is that he is back on the grass and that was nice to see yesterday.
"He's out there training on his own with the physios but that's a big positive. He's getting closer and closer.
"Ryan has been down in London seeing some specialists. We are awaiting some of the results coming back up from that visit.
"We're obviously just trying to get some further investigations to get to the bottom of Ryan's problem, so we'll wait for the results of that."
“Katic is back on the grass, which is great to see.”
Captain Tavernier injured his knee in a Europa league clash with Royal Antwerp back in February, while Jack has not featured since suffering a recurrence of his lingering calf issue against Dundee United three days later.