Hugh Shaw, the UK government official overseeing the salvage operation, said preparations were being made to allow the Hawk to begin the journey to Malta.
From Malta the 17,000-tonne drilling rig is to be taken to a yard in Turkey to be broken up.
The structure ran aground at Dalmore on Lewis in a storm in early August while being towed to Malta.
Last week, Transocean Winner was floated on to the deck of the Hawk, a semi-submersible ship used for transporting large maritime structures over long distances.
The rig Transocean Winner is expected to be moved on a heavy lift ship from Broad Bay, Lewis, on Wednesday.