Green completed running drills on field in the closing days of Australia's 214-run win over South Africa in Durban, before the team travelled to Gqeberha on Wednesday.
Australia remain confident he will play a part in the series, but the 27-year-old's hopes of returning on Sunday appear remote with just a three-day build-up.
There is better news for Josh Inglis, who has returned to the Australian squad after fracturing his finger and has started having light hits while wearing a splint.
"Cam's done some physical ... he's started running and is progressing well," Australia's coach Andrew McDonald said.
"As I said before the first Test match, we're confident that he can play a part in the series at some point.
"So he's got a few things to go through in the next three or four days. He'll progress that, and on the back of a soft-tissue injury it can accelerate quite quickly.
"With Josh, it'll be more about the comfort with the splints and how that progresses. He'll have bat-versus-ball sessions and we'll get more information there."
Australia have already indicated Green will not bowl in this series, potentially enhancing Beau Webster's chances of keeping his spot in the XI.
Webster bowled both pace and spin during the Durban win, but is at risk of being squeezed out of the side in the battle for middle-order spots.
If Green is not fit for the second Test, it would almost certainly enhance Nic Maddinson's chances of keeping his spot.
Australia are yet to decide if they will return Matt Renshaw to the top of the order after his 190 and 82 in Durban, and view Maddinson and Renshaw as interchangeable at the top and No.5 depending on conditions.
But Maddinson's inclusion would allow Renshaw to remain in the middle order if the wicket is similar to that of Durban's, which called for a drawn-out, grinding Test match.
Maddinson's selection was highly criticised at the start of the first Test, particularly after he recorded a 33-ball duck in the first innings before his quickfire 48 in the second.
McDonald again insisted he felt Maddinson had been harshly judged by pundits, coming to the 34-year-old's defence after doing likewise in the lead-up to the first Test.
"There was a heavy critiquing of his first innings. He left the ball well. There was a lot on offer, we felt," McDonald said.
"Then he probably played at a ball that he was leaving for the first 30-odd deliveries (when he edged a ball back on to his stumps).
"And the method he was applying until his dismissal was the way to get through that scenario, so a mature player reads the game well.
"The second innings, the scenario was slightly different. Flatter wicket, we needed to put runs on the board quickly to put some time in the game.
"So I thought he played both scenarios quite well, albeit the first things didn't turn out the way that he or we would have liked."
