The 32-year-old was forced off in the second half of Chelsea's 3-1 win at Norwich City on Sunday but Blues boss Jose Mourinho had been confident the player would be OK for international duty.
However, scans on Monday proved unfavourable and England boss Roy Hodgson boss called up Arsenal's Kieran Gibbs as back-up.
Following Chelsea's last-gasp win at Carrow Road over the weekend, Mourinho said: "Cole had something in his ribs that we need to assess. I want England to do well, so I hope he's fine."
Hodgson's squad met up for the first time at St George's Park in Burton on Monday ahead of their first match in the vital World Cup qualifying double-header.
The news of Cole's absence comes on the back of the withdrawal of Arsenal winger Theo Walcott - who is out with a stomach problem.
Cole, who has won 105 caps for England, will remain at Stamford Bridge to undergo further treatment with his club and means a likely starting place for Everton's Leighton Baines at left-back.
Hodgson's men require victory over Montenegro at Wembley Stadium and Poland four days later to be assured of qualification for the World Cup next year in Rio de Janeiro.
England currently lead Group H with 16 points, one ahead of Ukraine and Montenegro and they will be desperate to avoid finishing second where they would need to play a two-legged play-off.