The 26-year-old Edinburgh back-rower suffered the injury in the first half of the EMC Test defeat to the world champions.
Scotland team doctor James Robson said: "Ross was seen and treated at the stadium yesterday by Graham Lawson, consultant orthopaedic surgeon.
"He has been reviewed today at Spire Murrayfield Hospital and will now be in the care of shoulder specialist Miss Julie McBirnie for further assessment, which will give us a better indication of his rehabilitation schedule."
Rennie was winning his 20th cap and was hurt in the build-up to New Zealand's first of six tries, which was scored by Israel Dagg.
He was replaced by David Denton, his Edinburgh team-mate, and the length of his absence is to be determined.
Scotland's next Test is against South Africa on Saturday, with the final fixture of the year on November 24 against Tonga in Aberdeen.
Scotland will next train on Tuesday, with soft tissue injuries suffered by props Geoff Cross (ribs) and Ryan Grant (back), replacement hooker Scott Lawson (shoulder), lock Jim Hamilton (dead leg), centres Matt Scott (shoulder) and Nick De Luca (calf) and wing Tim Visser (flank).
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