The Mayor of London said the so-called Southern Rail Link scheme could be transformative in enabling more people to reach the UK’s busiest airport.
The proposal would see a new track laid from Terminal 5 station down towards the existing South Western Railway network, joining it at Staines.
From there, trains would travel through Twickenham and Richmond - or via Hounslow and Brentford - to reach Clapham Junction and Waterloo.
A second branch is proposed to run from the airport to Virginia Water, enabling services out towards Surrey, Berkshire and Hampshire.
That second branch would also mean that services from those counties could potentially make use of the existing Elizabeth line infrastructure to travel beyond Heathrow and through to Paddington, including a stop at the HS2 station of Old Oak Common.
The mayor told Ms Pidgeon: “I welcome the useful role that a Southern Rail Link to Heathrow could play in supporting a shift to sustainable modes [of travel], while transforming access to Heathrow Airport for passengers and staff from south London, Surrey and Hampshire.
“I am aware that a number of factors coming together in recent months have increased the prospects of a scheme moving forward.
“The Greater London Authority and Transport for London maintain regular dialogue with a broad range of interested stakeholders as we look to resolve the issues which have obstructed progress to date, while ensuring that any scheme meets the needs of Londoners and does not impact the reliable operation of the railway network.”
The project is being proposed by Heathrow Southern Railway Limited (HSRL), a company proposing to build and own the infrastructure - the capital cost of which was estimated in 2016/17 to be around £1.3bn.
The company says that train companies could then pay them an access charge in order to run passenger services on the route, the framework for which would be regulated by the Office for Rail and Road.
In June, Baroness Vere of Norbiton - the Government’s transport spokesperson in the House of Lords - said: “Government remains committed to improving rail access to Heathrow airport and our ambition is that any Southern Access to Heathrow scheme is funded and delivered by the private sector.
“Proposals need to be developed in a way that does not require government funding and secures significant benefits. We continue to discuss options and engage with the promoters as they work to resolve these outstanding issues.”
In 2016, it was revealed that a vast ‘ghost’ train station - with a length of 33 London buses - lies beneath Heathrow Terminal 5.
A Heathrow spokesman said at the time that the space was built in anticipation of the HSRL scheme.
That same year, Feltham and Heston MP Seema Malhotra said it was “scandalous that the world’s busiest airport is not connected to south London… for want of a few kilometres of track”.
The only current rail access into the airport is provided on an east-west basis from central London, via the Piccadilly and Elizabeth lines, along with the Heathrow Express service.