Repairs to HMS Astute have been completed and the submarine is at its Faslane base on the Clyde preparing to set sail again.
Mr Luff refused to confirm exactly when HMS Astute would leave the base "for security reasons", but said in a written Commons answer: "The department anticipates that this will be in December."
HMS Astute was on sea trials last month when it became stuck off the coast of Skye and ended up marooned for several hours.
Mr Luff said the "lower rudder skeg" was damaged by the grounding, and further damage was caused when the submarine was being towed to deeper waters.
The cost of the repairs is still being calculated, a Navy spokesman said at the weekend, and naval chief Andy Coles has lost his command of the vessel.
The submarine weighs 7,800 tonnes, equivalent to nearly 1,000 double-decker buses, and is almost 100 metres (328ft) long.
Its Spearfish torpedoes and Tomahawk cruise missiles are capable of delivering pinpoint strikes from 2,000km (1,240 miles) with conventional weapons.
The submarine's nuclear reactor means that it will not need refuelling in its entire 25-year life and it makes its own air and water, enabling it to circumnavigate the globe without needing to surface.
Built by defence giant BAE Systems at Barrow-in-Furness, Cumbria, it is the first in a fleet of six which will replace the Trafalgar-class submarine.
Mr Luff said current plans indicated it would be ready for operational handover in 2012.
As the base port of all the Navy's submarines from 2016, Faslane will be home to the whole Astute class.
The accident on October 22 happened almost exactly 50 years after the UK's first nuclear submarine was launched. HMS Dreadnought was launched on October 21 1960 by the Queen.