The 24 crew, mostly from Eastern Europe and Asia, are still on board the Maersk Tigris, anchored off Bandar Abbas.
Maersk said it still did not know the reason why the Marshall Islands-flagged vessel was seized, and noted it had been in an international shipping lane.
But Iranian authorities noted that it had been in the country's waters.
Iranian media cited an official as saying the ship was seized based on a court order in connection with a complaint made by a private Iranian company about debts.
On Wednesday, Maersk said it had been in contact with Singapore-based Rickmers Shipmanagement, from whom it charters the Maersk Tigris, and was "pleased to learn that the crew is safe and under the circumstances in good spirits".
The crew includes a British national, but no Americans, as initial reports suggested.
"We are continuing our efforts to obtain more information about the Iranian authorities' seizure - in international waters - of Maersk Tigris. We are not able at this point to establish or confirm the reason behind the seizure," Maersk added.
"We remain in close dialogue with the Danish ministry of foreign affairs."
A spokesman for Rickmers Shipmanagement said the vessel had been carrying commercial goods and no "special cargo" such as military equipment.
The incident has prompted the US Navy to dispatch a destroyer and a reconnaissance plane to the area to "monitor" the situation.
A general distress call sent out by the Maersk Tigris on Tuesday morning was picked up by US Naval Forces Central Command (NAVCENT).
A Pentagon spokesman said Iranian Revolutionary Guards Navy patrol vessels had approached the ship while it was moving through Iranian territorial waters, but inside an area that under international agreement is open to foreign ships making an "innocent passage".
The patrol vessels ordered the container ship to go deeper into Iranian waters, but the master refused. Only when they fired warning shots across its bow, did he comply.
The Pentagon spokesman called the actions of the patrol vessels "inappropriate".
The crew of a container ship seized by Iran in the Strait of Hormuz on Tuesday are safe and "in good spirits", Danish shipping company Maersk says.