The chief executive of the company implementing the HS2 railway project has said she believes Labour is still strongly supportive of the scheme.
Alison Munro acknowledged that HS2 would not be viable unless there was political consensus about its merits.
Senior Labour figures have voiced concerns about HS2's rising budget.
But former Labour transport secretary Lord Adonis warned his party not to give in to the "intense" temptation to cancel the scheme.
That would be an "act of national self-mutilation", he wrote, in an article for the New Statesman.
The HS2 scheme entails building a new high-speed rail link from London to Birmingham and to Manchester and Leeds.
The project's initial phase was approved in principle under the previous Labour government, and the scheme has had the backing of all three main party leaders since its conception - despite strong opposition among some backbench MPs.
'No blank cheque'
Chancellor George Osborne has predicted the high-speed rail network will be an "engine for growth" for the north of England and the Midlands.
Deputy Prime Minister and Lib Dem leader Nick Clegg has suggested the scheme could help to "heal the north-south divide".
But on Friday, former Labour chancellor and transport secretary Alistair Darling said he was withdrawing his support for it.
He warned of a potential "nightmare" for the rest of the rail network as a rise in HS2's budget from £32bn to £42.6bn would drain cash from other lines.
Another former Labour cabinet minister, Lord Mandelson, has warned the scheme could prove to be "an expensive mistake".
Asked about Mr Darling's comments, Ms Munro told BBC Radio 4's Today programme: "I should point out that Maria Eagle, who is the Labour transport spokesperson, yesterday confirmed her very strong support for HS2."
Opposition leader Ed Miliband has also said he is a "supporter" of the HS2 rail project but it should be scrutinised for "value for money".
But shadow chancellor Ed Balls told the BBC on Friday that he was concerned that the budget had gone "up and up".
He said: "We have consistently supported plans for a north-south rail link but it's got to work, it's got to be value for money."
A future Labour government would not offer the scheme a "blank cheque", he added.
But Ms Munro responded: "We are not asking for a blank cheque and we don't have a blank cheque."
She added: "I'd really like to knock on the head this talk of costs spiralling out of control. That is simply not true."
Ms Munro went on: "The current budget is £42.6bn for the infrastructure and £7.5bn for the rolling stock.
"We are absolutely determined, and the government is too, now to manage the project within that cost envelope."
The new budget included a £14bn contingency fund, she said, but HS2 Ltd was "determined" not to use it.
Supporters of HS2 argue that aside from shorter journey times, the main argument in favour of the project is the need to greatly increase passenger capacity.
Lord Adonis said the scheme was "going through the classic 'cold feet' period which bedevils every major British infrastructure project and which, with our short-termist political culture and poor project management, often leads to them being cancelled".
He continued: "This phase will continue until the 2015 election, when the temptation for Labour to claim it is 'saving' £42bn by proposing to cancel a 'Tory' project will be intense."
The peer warned Mr Miliband not to repeat the mistakes of the Labour government in 1974 which cancelled both the Channel Tunnel and a new London airport in the Thames Estuary.
"They were dubbed 'Tory extravagance' although, like HS2, their origins lay in the previous Labour government and there was nothing remotely right-wing about them," he said.
"It would be a similar act of national self-mutilation to cancel HS2 in 2015, six years into the project."
Construction on the London-West Midlands phase is expected to begin around 2017, once Parliament has approved the necessary powers - probably in 2015.
The onward legs to Manchester and Leeds could start being built in the middle of the next decade, with the line open by 2032-33.