The Chelsea defender also sat out Friday's 4-1 win over Montenegro due to the same problem, and will be unavailable for the visit of Waldemar Fornalik's side after failing to recover in time.
Leighton Baines - who filled in for Cole against Montenegro - will be expected to continue in his stead and claim his 21st England cap in the process.
Cole is joined on the sidelines by Tottenham right back Kyle Walker, who is unavailable due to suspension.
Manchester United duo Phil Jones and Chris Smalling are in contention to replace Walker, as England looks to ensure its place at next year's World Cup finals in Brazil.
Roy Hodgson's men will secure qualification with three points against Poland, which has already been eliminated following Friday's 1-0 loss to Ukraine.
England is one point ahead of Ukraine - which meets winless San Marino in its final game - heading into the encounter, but knows that failure to take maximum points will likely see the team face a playoff to qualify.