The Transnet National Ports Authority (TNPA) has confirmed the grounding of the vessels.
Videos online show tugs from the port trying to rescue the MSC Ines as it blocks the Durban harbour mouth.
A second vessel has reportedly broken free of its mooring and is stuck on a sandbank within the harbour.
The Port of Durban has suspended all vessel movements due to the weather.
TNPA said in a statement: “The inclement weather is also expected to disrupt operations at the Port of Richards Bay. Transnet’s marine services team and tugboats are assisting with several emergencies.
“Other vessels have broken mooring lines and are drifting in the channel. Whilst we are still in the process of determining damage to port infrastructure. Several TNPA premises have been flooded including the Transnet Maritime School of Excellence in Bayhead Road.
“We have assured all stakeholders of our urgent attention and allocation of resources to address these challenges. We will continue to monitor the situation closely via Port Control and the Harbour Masters’ offices, and will keep stakeholders updated.”