The study, commissioned by the UK and Scottish Governments and carried out by High Speed 2 Ltd, looked at broad options for improving rail capacity and journey times to the north of England and Scotland.
It said a continuous new high speed route from the northern end of the second phase of HS2 to Scotland would have the potential to reduce journey times from London Euston to around two and half hours.
The study looked at four route options. Route A would run to the west, following the West Coast Main Line, M6, and A74M corridor which splits near Carstairs to serve both Edinburgh and Glasgow.
Route B would follow the A1M to Newcastle and the A697 corridor to the border, where it runs through the Lammermuir Hill, to approach Edinburgh from the east.
Route C would follow the East Coast Main Line corridor east of Newcastle and along the coast to Edinburgh.
Route D would follow the A1M to Newcastle and through the Southern Uplands, where it splits to serve Glasgow and Edinburgh.
The report said the route to the west would have to pass through "areas of particularly undulating terrain in Cumbria and again in the Southern Uplands north of Lockerbie, requiring significant lengths of deep cutting, embankment and tunnels to fit a high speed alignment through the hills".
Routes to the east would have to "negotiate undulating ground as it passes through the Lammermuir Hills to the north of Berwick", and would also "encounter extensive areas of former mining activity".
The study said: "High speed routes would be feasible, but there are a number of challenges that would have to be addressed at a further stage of development in terms of environmental mitigation and management of cost and risk".
The report also looked at other options, including upgrades to existing routes.
However, it stopped short of backing a preferred option or route, saying that the options are at an early stage of design for feasibility assessment.
Scottish government infrastructure secretary Keith Brown said the report was a major milestone in the campaign to deliver high speed rail to Scotland.
He 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."