Pope will take to the field as a substitute fielder with the gloves, after the under-fire Buttler injured his left index finger earlier in the match.
England have also called Sydney Thunder 'keeper Sam Billings to their Test squad, with the miserably out-of-form Buttler in doubt for the fifth Ashes Test in Hobart.
England's keeping issues were further complicated by Bairstow's situation.
The right-hander provided their brightest spark of the tour on Friday when he went to stumps at 103no, but he did so with an injured thumb after being hit by Pat Cummins.
He was able to resume batting on Saturday morning after overnight scans, with English medical staff to assess his fitness at the end of the match.
The signs are looking good though, judging by the way the 32-year-old was able to battle through the pain on day three.
Bairstow added there was no way he was going to leave the field because of his injured thumb.
"Ultimately you're out there playing in an Ashes Test match, an Ashes New Year's Test match on pink day at Sydney in front of a crowd," he said after play.
"It's going to take a lot to get you away from that."
Still, it will stop him from keeping at the SCG.
Pope has kept at under-19 level for England and in Sydney grade cricket, but has not previously done so at Test level.
Stokes, meanwhile, continues to battle a side strain after being hurt while bowling on day two.
The star allrounder immediately left the field after clutching at his left side and while he later returned to the field he did not bowl again.
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