The 50m Eihatsu Maru ran aground in thick fog at 05:15 on Saturday, with 28 Taiwanese crew and a dog on board.
Solomons-Johannes said 19 of them were evacuated. The remaining crew and the captain are still on board as per the requirements of international maritime law. The dog remained on board on the insistence of the captain, its owner.
Solomons-Johannes said marine engineers had initially considered fitting a metal plate to the ship to tow it. However, the plan was aborted.
There is reportedly 90 tons of fuel and ammonia on board.
The Disaster Risk Management Centre conducted an aerial surveillance on Sunday afternoon to prepare for any eventuality.
Solomons-Johannes said the salvage operation had been suspended for the day at 16:45 due to unfavourable surf conditions that would have endangered the safe operation of the tug boats and the crew on-board.
It was not known what made the trawler run aground. Its engines remained in working order and generators continued to work.
The thick fog was suspected to have played a role.