Clarke's setback follows ankle problems which have limited the outside back to three tests this year.
"It’s unlucky for him," All Blacks assistant coach Jason Holland told New Zealand media.
"Caleb is a loss, and we’ve got a couple of big decisions to be made in the next couple of days about how we get the best guys on the park to start and finish the game."
All Blacks captain and lock Scott Barrett is expected to be back for the England clash, though, after missing the Murrayfield win with leg lacerations.
Beating England will keep the All Blacks on track to complete a "Grand Slam" of wins over the home nations for the first time since 2010.
England had a convincing 38-18 win over Fiji over the weekend after humbling Australia 25-7.
After England, Scott Robertson's New Zealand finish off the season-ending tour against struggling Wales.
Barrett aside, Holland said the players were "a bit beaten-up with bangs and knocks" after a long season but all were available for England.