Varney sustained the injury in The Railwaymen's shock 1-0 FA Cup exit to Conference side Telford United at Gresty Road last Saturday.
He dislocated his shoulder in a fall after being on the receiving end of a late tackle by Telford's Matthew Clarke in the first half of the match.
It was decided that the 21-year-old needs surgery now and the operation will take place next week, with a return date to training set for mid-April.
Meanwhile, young defender Chris McCready will miss another three weeks of action after the youngster broke down in training this week with a persistent calf muscle problem.
