The loosehead sustained the damage in the win over France in Paris.
“He had more scans and reviews with the specialists and it’s good news, he doesn’t need surgery,” said Richard Cockerill, the Edinburgh coach.
“He’ll undergo a process of rehabilitation and we’re hoping that will be, best case, six to eight weeks, which is an optimistic guide to where he’ll be at.”
Sutherland enjoyed an impressive Six Nations to thrust himself into Lions contention but time is running out for the Hawick man.
Warren Gatland will name his squad for the Lions tour on May 6 and they will play Japan in Edinburgh on June 26 before flying out to South Africa.
Scotland’s summer tour plans are yet to be revealed.