While initial proposals for the line only extend as far as the West Midlands, the new trains will also be able to run on conventional track as well.
Train journeys would be similar to those on Eurostar before the opening of the high speed link between St Pancras and the Channel Tunnel, where passengers travelled on conventional – or "classic'' – track through Kent, before switching to the faster line in France.
What is seen as the "one train two tracks" solution would bring some benefit to many major cities which would otherwise have to wait decades for their own dedicated high speed link.
This would mean that cities in the north would see journey times being cut by 36 minutes, because of the ability to use the new section of high speed track which is expected to be built by 2025.
The new line itself will bring the journey time from Birmingham to London down to 46 minutes, compared to the current 1 hour and 22 minutes.
But it would also mean that Liverpool could be reached in 1 hour and 32 minutes, rather than two hours and eight currently; Manchester in one hour and 31 minutes rather than two hours and seven minutes.
Glasgow would be three hours and 55 minutes away from London rather than just over four and a half hours, while the journey time to Edinburgh would come down to three hours and 37 minutes from four hours and 13.
"I would be very keen to see London to Scotland high speed trains from the outset of a high speed line," Lord Adonis, Transport Secretary, told the Daily Telegraph.
"There will still be significant reductions in journey time, even if a substantial proportion of the journey was made on the existing network."
Leaders from Birmingham, Bristol, Cardiff, Edinburgh, Glasgow, Leeds, Liverpool, Manchester, Newcastle, Nottingham and Sheffield are pressing the Government for a high speed line, which they say is essential for the regeneration of their economies.
The route of the a high speed line from London to the West Midlands is due to be unveiled at the end of the year. At the same time the company set up by the Government to handle the project is due to put forward recommendations where the line should go after that.
“A high speed rail link in the UK would mean shorter journey times, improved reliability, accessibility and greater capacity and all this, most importantly, would have a dramatic effect on the nation’s economic output," said Richard Leese, the leader of Manchester Council.
At the same time the Government is keen to link Britain to the European high speed network, with the vision of direct services to Paris and Brussels being operated from some provincial cities on both the new high speed line along with trains still using conventional track.
This would entail using a track loop at Stratford, east London which would link the British rail network with the current high speed line to the Channel tunnel, Lord Adonis said.
"It is one of our aspirations to have trains coming from Paris and Brussels to cities north of London. I hope we will see our cities linked directly to the continent."