Liberal Democrat Keith Aspden, who took over the leadership of the authority after the local elections in May, said the high speed rail project was "far from perfect" but said "that does not mean the project should be cancelled in its entirety.”
The current route of HS2, which is scheduled to arrive in the North in 2033, goes into Leeds and connects to the East Coast Main Line at Church Fenton, between Leeds and York in the Selby district.
It means that the high speed route will be able to serve York, cutting journey times to London from 111 minutes to 84 and from 112 to 57 minutes for trains to Birmingham.
Liberal Democrats councillors in York are calling on the Government to ensure that investment in the North’s rail infrastructure is not scrapped.
Coun Aspden said: “The news that Government is considering scrapping HS2 is concerning for the people who live and work in our city and the North of England as a whole.
"HS2 is far from perfect. It must be more transparent and accountable to the public, but that does not mean the project should be cancelled in its entirety.”
“At an unprecedented time for our country, building better infrastructure is critical for York’s and the North of England’s future development and creating more capacity on our railways is essential to achieving this.”
Separately Judith Blake, the leader of Leeds City Council, said the government announcement that it will look into the future of HS2 was “alarming”.
Terms of reference published by Transport Secretary Grant Shapps, said an independent review would look into the high speed rail project, and consider “whether and how [it] should proceed”.
Coun Blake, who also chairs the Core Cities UK partnership, said she it would be an “unmitigated disaster” if the report leads to the Government taking the option of finishing the project at the end of phase one – which connects London and Birmingham.
The next phase of the scheme – which would connect London with Leeds and Sheffield – is due to arrive by 2033, but has not yet been rubber-stamped by Parliament.
Coun Blake said: “What we need is to get on with the job and start building HS2 so its benefits can be realised as soon as possible and we can relieve the pressure on our creaking Victorian-era rail network.
“The terms of reference announced today are alarming, with one option being that we just build Phase one. This would be an unmitigated disaster for several of our cities that are attracting investment on the back of the promise of a high speed rail link. It would also endanger one of HS2’s key selling points, to better link up Northern and Midlands cities.
“The evidence for HS2’s economic and social benefits is irrefutable as far as we are concerned, it will help re-balance the UK, pump millions of pounds of investment into our local economies and is already supporting 9,000 jobs. It is the largest infrastructure project in Europe and should be seen as a statement of UK ambition that will pay for itself many times over in terms of increased productivity.
“Core Cities UK will join others and submit evidence to the latest review, but the terms of reference are flawed and call into question the Government’s commitment to creating a balanced, ambitious UK boasting the world class infrastructure our citizens deserve.”