The proposals would see the construction of a new Tube station at Battersea Power Station along with an intermediate station at Wandsworth Road in the Nine Elms area. The extension would reduce travel time from the West End to Battersea to around 15 minutes.
TfL said that up to 25,000 jobs and 16,000 new homes could be created as a result of the extension. It said that the extension would be "essential" to support the transformation of Nine Elms and Vauxhall.
Chancellor George Osborne announced a £1 billion funding package for the project in the 2012 Autumn Statement. Under the terms of the package, a public sector loan will finance the construction of the extension with a Government repayment guarantee to minimise borrowing costs.
Loan repayments will then be made partly from developer contributions under section 106 agreements or community infrastructure levies and partly from revenue collected from business rates within the Nine Elms Enterprise Zone.
"South London is one step closer to a possible new Tube extension," said TfL managing director of planning Michèle Dix in a statement.
"Extensive public consultation has shown that the majority of local residents and businesses are behind this proposed Tube extension, which will create access to the Underground for thousands of people, as well as cutting journey times from the Battersea area to the West End and the City to about 15 minutes," he said.
News of the application for the extension, which will run through the three London Boroughs of Lambeth, Southwark and Wandsworth, was welcomed by Wandsworth Council leader Ravi Govindia.
"Bringing the Tube to Battersea has long been the ambition of this council and we are now within touching distance," he said in a statement.
The Secretary of State for Transport will now decide whether a public inquiry is required. This would likely be held in the autumn and the Government would be expected to make a decision in the autumn 2014. If approved, construction could start in the spring 2015 and the new stations could open in 2020.
