The government said it would take over the franchise when this missed payment occurred - though this may not be for some months.
The franchise would then be put out for tender to a new buyer from late next year.
Existing operational staff would transfer to a new state company which has been set up to operate the route.
Separately, Virgin Trains told the BBC's You and Yours programme that it would bid for the East Coast service.
Virgin said it wanted to operate the service "in principle", depending on the details of the new contract.
National Express won the franchise in 2007, after it was surrendered by GNER. It agreed to pay the government £1.4bn to run the East Coast main line, which runs between Edinburgh and London, until 2015.
At the time, many rail analysts said it had paid too much for the franchise.
In a trading statement, the company said that the "challenging economic environment" meant it was seeing fewer passengers on the East Coast main line and "significant" levels of people downgrading from first-class and full fares.
National Express indicated that it would walk away from the loss-making route, and not put further money into it, after failing to alter the terms of its franchise agreement.
The firm's chief operating officer, Ray O'Toole, later told the BBC it was possible that it would continue to operate the route if economic conditions improved.
However, BBC transport correspondent Tom Symonds said it was generally accepted that the franchise would return to the government at some point after National Express failed to alter the terms of its agreement.
Lord Adonis told the BBC: "The government is not prepared to renegotiate rail franchises, because I'm simply not prepared to bail out companies that are unable to meet their commitments.
National Express chief operating officer Ray O'Toole said he wanted to ensure passengers were not disrupted
"It is simply unacceptable to reap the benefits of contracts when times are good, only to walk away from them when times become more challenging."
In a statement, the government added that it believed it also had grounds to end National Express's two other rail franchises - East Anglia and c2c.
But the company said it felt the government had "no grounds" to do this and would challenge any such attempt in court.
Shadow transport secretary Theresa Villiers said the move showed the extent of Labour's "failure and incompetence" in running the rail franchising system.
"This Labour government has used the franchising process to push the train operators to make wildly optimistic bids, which has meant squeezing passengers for higher and higher fares. Now the chickens are coming home to roost," she said.
Liberal Democrat transport spokesman Norman Baker said that National Express should not be allowed to run any other train services.
"It doesn't seem to me right that the taxpayer should pick up the loss from one and leave the profits from the other two with National Express," he told the BBC.
He added that leaving the East Coast line in public hands would provide a comparator which would be "useful in driving up standards across the whole industry".
The transport union, the RMT, welcomed the development and called for the franchise to remain in government hands permanently.
"It should be a long-term solution to the chaos that privatisation has brought to the UK's most lucrative rail franchise," said RMT general secretary Bob Crow.
National Express also announced that its chief executive, Richard Bowker, the former Strategic Rail Authority (SRA) chief, would be leaving the firm to become boss of Union Railway in the United Arab Emirates.
It denied that his departure was owing to financial problems at National Express.
The company is trying to reduce a debt pile that stood at nearly £1.2bn at the end of 2008.
Cuts have been made in dividend payouts to shareholders, while 750 jobs have also been lost.
Last month, National Express started charging passengers for reserving a seat on the East Coast and East Anglia franchises, while it has also removed its online ticket sale discounts.
Earlier this week, rival transport company FirstGroup said a takeover approach it had made for National Express had been rejected.
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