Warner picked up the injury while training in the nets, struck on the hand by Mitchell Johnson. If he does not recover in time, it means Ed Cowan is likely to open with Shane Watson, who would otherwise bat lower down, despite preferring the top role.
Meanwhile, skipper Michael Clarke has made a good recovery after his hamstring niggle and is set to play. He didn't feature in the warm-up against India A though, while Watson made 84 in the drawn match.
Chief selector John Inverarity said on ABC Radio: "David's thumb has improved in the last couple of days, he wasn't fit enough to play in this game, he had some throw-downs yesterday with a cricket ball, and we're hoping his fitness improves a great deal in the next couple of days so he comes into contention.
"We won't make a decision on that until a few days out from the Test.
"There is some concern with David because he hasn't been able to practice, but Michael had an hour's batting in the nets yesterday and he played beautifully, so I'd say there are very few concerns around Michael because of his experience and the fact he was able to practice."
Looking towards the bowling side of things, Inverarity hinted that while the Chennai deck would be suited to a spinner or two, they would likely still go with a seam attack, adding a spinner and an all-rounder.
He added: "The wicket for the Test match is already prepared. We could've played on it today, we could have played on it yesterday or the day before.
"There is not one blade of grass, so you'd suggest it is a wicket for spinners. But with our Australian team I think the strength lies more with the pace bowlers than with the spinners.
"We've got the option of playing two pace bowlers and two spinners, but it'd more likely that we'd go in with three pace bowlers and one spinner, plus some spin support. I don't think it's clouded. We know what we're going to do.
"The Indian players, of course, are very good players of spin. Yesterday, our spinners didn't bowl well and the Indian batsmen took heavy toll of them, but we're fairly clear in regards what we're going to do in terms of spin bowling in the first Test match."