The Ever Given, a Panama-flagged ship that carries cargo between Asia and Europe, ran aground on Tuesday morning in the narrow canal following strong winds.
Shipping journal Lloyd's List estimated that each day the canal is closed disrupts £6.6 billion of goods that should be passing through the waterway.
The ship's Japanese owner, Shoei Kisen Kaisha, offered a written apology on Thursday, saying: "We are determined to keep on working hard to resolve this situation as soon as possible.
"We would like to apologise to all parties affected by this incident, including the ships travelling and planning to travel through Suez Canal."
The Ever Given had two pilots from Egypt's canal authority aboard to guide it when it ran aground.
Bernhard Schulte Shipmanagement, the company that manages the Ever Given, said the ship's 25-member crew were safe and accounted for.
Authorities began work again to free the vessel on Thursday morning after halting for the night, an Egyptian canal authority official said.
The source said workers hoped to avoid offloading containers from the vessel as it would take days.
So far, dredgers have tried to clear silt around the massive ship.
Canal service provider Leth Agencies said at least 150 ships were waiting for the Ever Given to be cleared.
Vessels in the ‘queue’ include those near Port Said in the Mediterranean Sea, Port Suez in the Red Sea and those already stuck in the canal system on Egypt's Great Bitter Lake.
Cargo ships already behind the Ever Given in the canal will be reversed south to Port Suez to free the channel, Leth Agencies said.
Authorities hope to do the same to the Ever Given when they can free it.
Egyptian forecasters said high winds and a sandstorm plagued the area on Tuesday, with winds gusting as high as 30mph.
An initial report suggested the ship suffered a power blackout before the incident, something Bernhard Schulte Shipmanagement denied on Thursday.
"Initial investigations rule out any mechanical or engine failure as a cause of the grounding," the company said.