Morkel's injury adds to a long list of injured South African fast bowlers, with Dale Steyn, Vernon Philander and Chris Morris already out of action for extended periods.
Moosajee was speaking shortly before South Africa wrapped up victory in the first Test against Bangladesh, the first of a scheduled 10 Tests during the home season.
He said Morkel would have scans today but a preliminary assessment was that he was likely to need six weeks of rest.