A crack developed on the upper deck of the Genmar Companion in the middle of December as the vessel was travelling 40 miles west of Tory Island, Co Donegal.
It was sailing from Rotterdam to New York to deliver its cargo of 54,000 tonnes of vacuum gas oil when the crack was discovered. The ship has been sheltering off the Copeland Islands since 16 December.
Environment Minister Alex Attwood said: "I am pleased that the ship to ship transfer is now imminent and that no risks were taken which might have endangered the marine environment during the high winds.
"I am continuing to keep a close eye on the situation and one of my officials boarded the receiving vessel with MCA colleagues just prior to the berthing operation," he said.
The next 24-36 hours will be critical in ensuring that no polluting substances enter the sea.
Environment Minister Alex Attwood
BW Seine, the offload tanker, and the Genmar Companion are being held side-by-side 1.5 miles out to sea in Belfast Lough. The cargo is being transferred from one to the other using specialist hoses.
There is a one mile exclusion zone around the vessels to give the tugs a safe working area.
Ken Fleming, who is an Irish inspector for the International Transport Workers Federation, confirmed there is a two-year-old girl on board the stricken vessel.
"It's not unusual from time to time for families to travel with some of the officer members onboard the vessel," he said, before adding that this was considered a perk for senior staff.
He told UTV there had been "ample time" time to remove the staff from the vessel to a safer place.
He said only crew members, for example with emergency training, should be on the vessel and family members should have been removed, but added that the rescue "wouldn't be as dramatic" as needing a helicopter.
The transfer process was due to begin last weekend, however, poor weather conditions delayed the offload tanker's journey to Northern Ireland. It was then expected to commence on Monday, but high winds delayed the process once again.
Since then, stormy conditions have prevented the transfer from taking place. However Mr Fleming said there has not been consistently inclement weather since the middle of December and the family on board should be "removed without haste".
"The environment on board the vessel is not conducive to a two-year-old baby.
"It wasn't constructed in such a way that it would provide amenities that would make it suitable for those on board.
"I or someone like me should be given access to have an independent assessment of how the lads are feeling on board, given what they've been through," he added.
The operation is expected to take approximately 24-36 hours and on completion, the Genmar Companion will enter Belfast Harbour for repairs.
Alan Chambers, an Independent councillor for North Down, has repeatedly voiced his concern regarding the possibility of pollution during the transfer.
He told UTV he is worried about any possible environmental impact the maritime operation may have on the stretch of coastline, which is an area of scientific interest.
"The Copeland islands are home to a great deal of bird life so any form of pollution would be a total disaster for this area.
"My main concern is that this whole thing started in Rotterdam and culminated in the ship getting into distress off the coast of Ireland in the Atlantic Ocean and suddenly it has appeared here two miles off the North Down coast.
"I just don't believe it is our problem but it has suddenly become our problem."