Vince Cable was forced into an embarrassing apology last night after being secretly recorded saying he could "bring the government down" by walking out of the coalition if "pushed too far" in negotiations with Tory ministers.

The Lib Dem business secretary was taped by journalists posing as constituents saying that being in the coalition was "like fighting a war" and that he could use the "nuclear option" of quitting. He also told the Daily Telegraph undercover reporters, who were posing as mothers worried about changes to benefits, that he believed David Cameron wanted to reduce or scrap the winter fuel allowance for pensioners.

Cable's comments were out of kilter with claims made today by Danny Alexander, the Lib Dem chief secretary to the Treasury, who stood full square with the Conservatives as he insisted in an interview with the Financial Times that the coalition would "stand firm" in implementing its £81bn of cuts next year, and claimed that a "personal chemistry" at the top has proved a powerful factor in sealing the coalition together.

He said that some of the things the government was having to do were "bloody hard" and would require "a lot of resilience from all parts of the coalition".

But Cable showed unease as he criticised the speed at which the coalition was attempting change, saying the government was trying to do "too many things" in terms of reforms – a lot of which he said were "Tory inspired" and had not been thought through. "We should be putting a brake on it," the business secretary said.

The gaffe came three weeks after Cable caused a furore by threatening to abstain in the vote on tuition fees, despite being the minister responsible for universities.

Cable is the highest-ranking Lib Dem member of the coalition after Nick Clegg, the deputy prime minister. He said last night he was "embarrassed" by his remarks but had no intention of resigning. Downing Street declined to comment on Cable's remarks last night.

But the indiscretion raises a fresh question over whether the coalition can manage to survive intact until the planned date of the next general election in May 2015 – and whether Cable can hang on to a top job for much longer.

It is well known in Westminster that many cabinet members would happily see the business secretary returned to the backbenches, and would consider David Laws as his worthy replacement. Laws quit as chief secretary to the Treasury earlier this year after it emerged he had broken parliamentary expenses rules by renting a room from his lover.

In his conversation with the undercover reporters at a surgery in his Twickenham constituency, Cable said he had been involved in a "big argument" over how to deal with the banks, with the Lib Dems pressing for a "very tough approach" that was opposed by "our Conservative friends".

Asking the reporters not to quote him "outside", Cable said that while he had to pick his fights carefully, he was prepared if necessary to quit the coalition.

"I have a nuclear option; it's like fighting a war. They know I have nuclear weapons, but I don't have any conventional weapons. If they push me too far, then I can walk out and bring the government down; and they know that.

"So it is a question of how you use that intelligently, without getting involved in a war that destroys all of us. That is quite a difficult position to be in, and I'm picking my fights. Some of which you may have seen."

Cable pointed to the scrapping of child benefit for higher earners, which he said had been done in a "rather cack-handed way", creating "a certain amount of injustice", for which he blamed the Conservatives. But it is his suggestion that the winter fuel allowance could also be under threat (despite repeated assurances by Cameron it will be protected) which may cause greatest consternation in No 10. "They haven't yet done the winter fuel payments, but that's coming, I think," Cable told the reporters.

Last night some of Cable's colleagues were incredulous he had been outspoken to two alleged constituents whom he had never met previously. It was being suggested that he should have made a basic check to ensure they lived in the constituency. His colleagues feared the incident would weaken him politically, increasing the chance he would resign and be replaced by a more rightwing figure such as Laws, undermining the balance of the coalition.

Labour MP Tom Watson said that Cable's comments would "strike fear into millions of pensioners battling through the coldest winter in decades". John Denham, the shadow business secretary, said Cable's remarks showed the coalition was "paralysed" by cross-party squabbles.

"This infighting explains why the Tory-led government has three times abandoned plans to publish a plan for growth. The government is paralysed while millions wonder if their job will go next," he said.

Cable was also recorded referring to Howard Flight, the disgraced Tory peer who said child benefit changes would encourage the poor to "breed", as "a nice but rather silly public schoolboy with a few prejudices to boot".

In a statement last night, Cable said he remained committed to the coalition and had no intention of leaving.

"Naturally I am embarrassed by these comments, and I regret them," he said. "I have no intention of leaving the government. I am proud of what it is achieving and will continue to play my full part in delivering the priorities I and my party believe in, which are enshrined in the coalition agreement."

A Labour party spokesman added: "Vince Cable's unguarded words show this really is a Tory-led government. Under David Cameron's premiership, the Lib Dems have broken promises on VAT, tuition fees and police numbers and got little in return."

Alexander appeared more upbeat about the Lib Dems' alliance with the Conservatives in his FT interview, published today.

The Lib Dem minister behind Britain's austerity programme vowed to resist pressure to spend more money next year to lessen the pain of the cuts, even if the recovery started to falter.

"The plan is right," he said. "The government is absolutely going to stick to it. People should be in no doubt at all.

"The most important thing is for the government to see the plan through. That's what is going to establish confidence in the economy and the credibility we need going forward."