The Sun originally reported that HS2 trains could stop at a new hub at Old Oak Common in West London's suburbs due to soaring inflation, which would either delay the redevelopment of Euston station until 2038 or see it face the axe completely.
But Hunt said there were no "conceivable circumstances" where the project would have its terminus changed when asked by BBC News.
Hunt acknowledged that the UK does not have "a good record" of delivering complex, expensive infrastructure quickly, but he is "incredibly proud" that HS2 is being built under a Conservative government.
"We're going to make it happen," he added.
In April 2020, the Department for Transport released a 'full business case' for HS2 that stated that the target timeframe for services launching between Old Oak Common and Birmingham was 2029-2033, whereas for trains between Euston and north-west England the range was 2031-2036.
"Euston is an important part of realising the benefits of HS2 and that work should continue on the section from Old Oak Common to Euston," the document stated.
"Notwithstanding this, Euston is a very challenging, complex major programme and given its current status, Old Oak Common will be expected to operate as a temporary terminus for a period of time."
Despite the denials, deputy Labour leader Angela Rayner said reports of HS2 not being routed all the way to Euston was an example of the Conservative government failing to deliver on big infrastructure projects.
When asked about the possibility, she said: "I don't think it is a sensible idea for the government to continue to promise things and then not deliver them, especially long-term infrastructure projects.
"We know if you listen to business, if you listen to the experts that actually long-term infrastructure planning will boost our growth.
"And yet we've seen, with the Conservatives' 13 years of failure, growth has stagnated in the UK, we are lagging behind other European countries.
"We have to invest in our infrastructure for the long-term and, once again, the Conservatives are moving us from crisis to crisis.
"And actually with long-term projects like this, the more you change it, the more money it costs, so we really do need to commit to HS2 funding and commit to infrastructure projects, like we have in the Labour Party, so we can put our economy on the long-term footing."
Last summer, a 2,000-tonne, custom-made tunnel-boring machine named Dorothy became the first HS2 machine to complete a tunnel breakthrough as part of the first of 64 miles of intricate tunnels that need to be completed for the Birmingham to London line.
