Mitchell suffered the injury during England's 73-8 win over the FIJI Water Flying Fijians in Liverpool last Saturday.
The 24-year-old Quirke recently signed with Newcastle Red Bulls after leaving Sale Sharks.
He has made two appearances for England and became the country's youngest scrum-half in more than 90 years when he made his debut in 2021.
Quirke also played for an England XV against
a France XV in a Nations Championship warm-up match last month.
Leicester's Jack van Poortvliet and Bath's Ben Spencer are the other scrum-half options for coach Steve Borthwick ahead of England's final Test of the summer against Argentina.
Meanwhile, in other games this weekend, the Flying Fijians will take on Scotland at Murrayfield, Edingburgh at 1.10am on Sunday, South Africa will play Wales at 3.40am, and Argentina will take on England at 7.10am.
On Saturday, New Zealand will go against Ireland at 7.10pm, Japan will play France at 8.40pm, and Australia will host Italy in Perth at 10.10pm.
Source: BBC News
