Up to 60 trains will be needed to transport passengers at around 225mph.
The contract, which is worth £2.75bn, will be awarded in 2019.
Advance warning about the start of the bidding process was published on Friday by HS2 Ltd, the organisation responsible for developing the £55.7 billion project.
Transport Secretary Chris Grayling described the search for a train manufacturer as “a major step” towards Britain getting a new railway which will carry more than 300,000 people each day.
Mr Grayling said building and maintaining the trains will be a “great opportunity for British-based businesses and suppliers” with the creation of hundreds of jobs.
He added: “We have held discussions with UK suppliers to make sure they are in the best possible position to win contracts.”
HS2 Ltd’s managing director of railway operations, Chris Rayner, said: “We are looking for a supplier who is capable of delivering and maintaining some of the world’s most advanced rolling stock, with designs putting the passenger at their heart to ensure seamless, accessible, fast and reliable journeys.”
Construction work on phase one of HS2 from London to Birmingham will start in the spring, subject to its parliamentary bill getting royal assent in the coming weeks.
Phase one is due to open in December 2026, with a second Y-shaped phase launching in two stages.
Phase 2a from the West Midlands to Crewe will open in 2027 and phase 2b, from Crewe to Manchester and from the West Midlands to Leeds, South Yorkshire and the East Midlands, will begin operation in 2033.
The Department for Transport said 25,000 jobs and 2,000 apprenticeships will be created during the construction of HS2.
Hitachi Rail Europe will bid for the contract to build the HS2 trains.
The rail manufacturer is head-quartered in London and builds trains at its plant in Newton Aycliffe, County Durham.
Its sales director, Nick Hughes, said: “Our HS2 trains would be built in Britain, for Britain, and combine world-leading Japanese Shinkansen technology with British manufacturing know-how.
“By building at Hitachi’s facility in the North East, our high speed trains would bring together values of pioneering engineering, industry-leading reliability and innovative design.
“The trains will offer next generation passenger experience, utilising the latest smart and digital technologies.”