Service suspended on Northern Line between High Barnet and Finchley Central due to signal failure at High Barnet.
The Mayor's office says that the Northern Line extension will reduce pressure on Vauxhall station and existing Northern Line services, in addition to boosting the area's economy.
It is expected that work on the line could begin by 2015, with services in operation by 2020.
A consultation on the plans is already underway.
The Chancellor has announced a £1 billion pound loan to London Underground to finance the extension of the Northern Line to Battersea.
Mayor of London Boris Johnson said:
“I am delighted that after months of intensive discussions and hard work we have got the go ahead from Government on financing an extension to the Northern Line, which is hugely significant as Nine Elms is one of the areas with the greatest prospect for new development in the whole of the capital.
"It will be an incredible confidence boost for developers preparing to invest there and it will also be a considerable boost for local people who will benefit from the new Tube link.”
The Chancellor has announced that the Northern Line will be extended to Battersea, as part of his Autumn Statement.
George Osborne also announced £1 billion of investment in the country's roads, including upgrades to the A1, the A30 and the M25.
Minor delays on Northern Line between High Barnet and Camden Town, and between Camden Town and Kennington due to signal failure at High Barnet, and at Leicester Square.
Underground station closed on Northern Line at Mill Hill East due to staff shortage.
Minor delays on Northern Line between Stockwell and Morden due to person ill on a train earlier at Clapham South.
Underground station closed on Northern Line and Bakerloo Line at Charing Cross due to fire alert.
On the tube, trains are not stopping on the Northern Line Northbound at Euston due to power failure. Tickets are being accepted on local bus routes.
Underground station closed on Northern Line at Hampstead due to fire alert.