McCaw, 33, will miss the Crusaders' final three regular-season clashes against the Hurricanes, Blues and Highlanders after sustaining the problem in the 3-0 series whitewash.
However, he is targeting a return in week one of the finals, starting on July 19, providing the Crusaders qualify.
The seven-time Super Rugby champs hold top spot in the New Zealand conference on 41 points, three clear of the Highlanders, with the Hurricanes third on 37.
All Blacks coach Steve Hansen confirmed on Tuesday that McCaw suffered rib damage in either the first or second Test against England and aggravated the injury in Saturday's 36-13 thrashing of the tourists.
"The doctor had a look at it, and they didn't think it was great - they saw it was badly bruised," Hansen told Radio Sport. "He played in the last one and he was really sore after that, so he's obviously done some more damage to it in the third Test."
McCaw's injury is unlikely to have any impact on his availability for the All Blacks' clash against Australia in Sydney on August 16.
New Zealand will break the tier-one record for consecutive Test victories with a win over the Wallabies, who have won their past seven matches including a 3-0 series whitewash against France.
New Zealand's winning streak currently stands at 17 games, with Hansen's side having not been beaten by the Wallabies since 2002.