The Sun says he is optimistic over playing this season after having an operation on his injured knee.
The City forward, 23, is refusing to set a time-frame on a recovery but surgery on his damaged anterior-cruciate ligament is understood to have been successful.
Sane went under the knife this week in Austria rather than use City's favoured surgeon Dr Ramon Cugat in Barcelona.
He went with Professor Christian Fink on the recommendation of his Germany team-mates.
Austria is now seen as one of the best places for ACL treatment after building a good reputation in snowboarding and skiing, where the injury is common.
Sane's decision was a personal choice and not linked to Bayern Munich chasing him this summer before injury struck.