The bodies want John Healy, minister for local government, to employ Tax Increment Financing (TIF) to fund the ВЈ850m project, according to Estates Gazette.
The letter says that the extension is crucial to the success of ambitious plans to regenerate more than 200 acres in the Battersea and Nine Elms opportunity area.
The tax model, which is common in the US, would involve councils selling bonds to forward fund infrastructure.
That money would then be repaid via an increase in business rates when the regeneration of an area was completed.
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Pilot schemes are expected to be announced by the government in the Pre Budget report this autumn.
Mayor of London Boris Johnson is also understood to have agreed to divert money that he had proposed to raise for Crossrail via a levy on commercial development in the area towards funding the Northern Line extension.