Both the players have been ruled out of the third and final Test against South Africa.
Green copped a blow on his finger by an Anrich Nortje bouncer and was forced to retire on Day 2, but he batted through pain on Day 3 to score an unbeaten half-century.
Surgeons are hopeful the 23-year-old should recover in time to be considered for selection for the upcoming Test tour of India, slated to get underway on February 9.
Another Australian cricketer to sustain a finger injury during the Boxing-Day Test was pace spearhead Mitchell Starc.
The left-arm fast bowler, who picked the injury while fielding, faces a potentially longer stay on the sidelines and will be in a race against time to prove his fitness for the Border-Gavaskar Trophy 2023.
The duo has been ruled out of the third and final Test against South Africa, starting January 4 at the Sydney Cricket Ground.
A report published in cricket.com.au on Thursday (December 29) stated that Green had consulted a specialist, who had recommended surgery.
On the other hand, Starc doesn’t need to undergo a surgery for the tendon damage to his left middle finger but he might not be available for the series opener against India in Nagpur.
"India's the next big tour and we'll see where the time frame is with that. It's my bowling hand so I've got to be pretty careful and make sure it heals properly," Starc was quoted as saying after win against the Proteas.
"The irony is (Green) will be back before I am. Bones are a bit quicker in the healing process, the tendon is a little bit different. I think we'll both be seeing the same specialist.
"I still think everyone's confident it will be more front-end (of India tour). It will obviously depend on how it heals and how quickly it can do what it needs to do," he added.
Despite being in pain, Mitchell Starc delivered 18 overs in South Africa’s second innings and claimed the wicket of Proteas opener Sarel Erwee.
"I wasn't sure what to expect. I need the middle finger for control more than anything," said Starc.
"I've had a lot of painkillers. I could have jabbed it (to numb the finger) but I feel like I need the feeling of the ball on finger, otherwise I feel like I would have been spraying it everywhere.
"I've played with a broken foot before… it's Test match cricket. I've been on both sides of being injured and then (bowling as part of an attack) with one man down. Having (Cameron) Greeny with a busted finger as well probably helped that decision to try to push through," he further remarked.
(With PTI inputs)