The Transport Department will instead announce a new rail plan on Thursday, involving £96bn of funding for new routes in the North and Midlands.
Sources said the impact of scrapping the Leeds leg of HS2 would make journeys longer by 20 minutes.
But the government is set to argue the new plans will deliver comparable benefits more quickly and cheaply.
High Speed 2 is a planned new high-speed railway line, originally meant to connect London with the city centres of Birmingham, Manchester and Leeds.
But Conservative MPs expressed concerns about the cost of the eastern leg connecting the West Midlands and Leeds, and there were rumours it would be scrapped.
According to a report in the Sunday Times this weekend, Transport Secretary Grant Shapps will announce two shorter high-speed routes created in part by upgrading existing lines. One will run between Leeds and Sheffield, another from Birmingham to East Midlands Parkway.
The government is also expected to put money aside to explore setting up a tram service for Leeds.
The Northern Powerhouse Partnership, a group of northern local authorities and business leaders, said the decision to scrap the Leeds leg of HS2 was a mistake.
Director Henri Murison said: “The reported loss of any of the new line on the eastern leg of HS2 is damaging, reducing the benefits of the section being built now between Birmingham and London.
“Without the benefits to areas such as Yorkshire and the North East, HS2’s status as a project to drive the whole of the UK is undermined considerably.”
A proposed Northern Powerhouse route from Leeds to Manchester is now expected to be made up of some new line, but it will mostly consist of upgrades to the existing track.
The new track on the route will not allow high-speed rail travel.
The route is not expected to go via Bradford, a key request of many in the city and surrounding area.
Naz Shah, the Labour MP for Bradford West, tweeted: “This is Boris pulling the whole damn rug from under our feet and ripping up the floor behind him!”
The upcoming rail plan matters to many thousands of people and businesses. Whether they’re taking a train for commuting, a business meeting or for leisure, what will their future options be? And how long will those journeys take?
Rumours that the eastern leg of HS2 could be scaled back or scrapped altogether have been swirling for months. It now looks likely that only part of the route linking the West Midlands with Yorkshire will get a new high-speed track.
The government’s view is that the alternative option for HS2 would still make journey times faster, but deliver benefits more quickly. There’s clearly a cost element too. But it doesn’t bring the same benefits when it comes to capacity.
Meanwhile, there’s disappointment in cities such as Bradford. They see the delivery of Northern Powerhouse Rail, including a new line from Leeds to Manchester via Bradford, as crucial to regeneration and creating opportunities for their residents.
We’ll learn on Thursday what the final plans are. At the moment it looks like they will receive significantly more of a welcome in the East Midlands than the North of England.
The shadow transport secretary, Jim McMahon, accused the government of trying to back out of promises made on badly-needed major infrastructure projects, and described the reported plans as “half-baked and repackaged”.
Conservative MP Kevin Hollinrake, who represents Thirsk and Malton in North Yorkshire, told the i newspaper the downsized plans indicated the government was “not willing to put our money where our mouth is”.