The Brexit secretary's comments come as ministers mull over whether or not to go ahead with the high-speed line - with a decision expected in the coming weeks.
Whitehall’s spending watchdog said last week HS2 is over budget and behind schedule, while the National Audit Office (NAO) warned that it is impossible to “estimate with certainty what the final cost could be”.
However Mr Barclay told BBC One’s The Andrew Marr Show the project was key to the Government’s plans to raise the economic performance of all parts of the country.
“We have a strong commitment to levelling up all parts of the United Kingdom. High-speed two is a key part of that – not just from speed but more from a capacity point of view in the line," he said.
“And that is a very clear commitment we have given the North.”
Asked if his gut feeling was that the project should go ahead, he replied: “Yes.”
Transport Secretary Grant Shapps has said the Government will make a decision on whether to go ahead with the project in “weeks rather than months”.
Political leaders in northern England and business groups claim HS2 is vital to boosting transport links across the region and providing increased capacity on the overcrowded rail network.
Construction firms warn that scrapping it would cause major damage to the industry.
However, opponents claim HS2 is too expensive and environmental groups have criticised it.