The governments agreed that there needs to be further work to bring train journeys down to three hours or less, while easing the severe congestion on cross-border routes.
The plan also hopes to create jobs, investment opportunities and economic and environmental benefits for Scotland.
Scottish government Infrastructure Secretary Keith Brown said: "High speed rail will bring billions of pounds worth of benefit to Scotland’s economy and an infrastructure project of this magnitude – possibly the biggest Scotland’s ever seen - means jobs, investment, benefits for the economy and benefits for the environment."
"This plan will bring to life our target of 3 hours or less Glasgow and Edinburgh to London train journeys, which will lead to a significant move from air to rail, bringing big reductions in carbon emissions," he added.
Work is now planned to be carried out in 2017 to identify options with the best business case for implementation from 2019 onwards.
The reduced journey time comes as HS2 gets set to improve journeys between Scotland and England. When Phase One opens in 2026, new HS2 trains will be arriving in Glasgow from London in just under four hours.
HS2 is estimated to bring £3bn of benefits to Scotland.