The leader of the Liberal Democrats, Sir Vince Cable, urged Michel Barnier to stop talking up a no-deal Brexit and instead make contingency plans for a second referendum after the EU’s chief negotiator admitted that a parliamentary impasse in the UK was standing in the way of a deal.

Cable said the government and Brussels had up until now been emphasising the danger of the UK crashing out in order to sell a “bad deal” to parliament.

But the former business secretary told Barnier during a meeting in Brussels that the so-called people’s vote was gathering support across the political spectrum, and the European commission needed to ready itself for the eventuality.

Emerging from an hour-long meeting, Cable said it was clear that the former French minister had a keen grip on the political situation in Westminster.

Barnier was said to have told Cable that he recognised that the major hurdle standing in the way of a deal was the impossible parliamentary arithmetic facing the prime minister.

“The UK needs to come up with something that meets their requirements,” Cable said of his conversation with Barnier. “It’s very clear really: it is a British problem rather than one on the European side.”

But Cable, who was meeting the EU official with the leader of the SNP in Westminster, Ian Blackford, and representatives of Plaid Cymru and the Greens, said the message to Barnier was that there was no way out of the logjam, and that the strength of support for a second referendum was growing daily.

Cable said: “My main message was that a people’s vote is a very live political option. I wanted to know that the commission is doing some contingency work. He said that he was the negotiator and this was a matter for the commission, and no doubt the message will be passed on. He obviously had an very clear understanding of what was happening in the UK.”

A joint statement from the British politicians attending the meeting said: “With sensible politicians from all parties uniting, we pointed out to Mr Barnier that there is a genuine cross-party consensus that our exit from the European Union must not be assumed.”

The president of the European Central Bank, Mario Draghi, said on Thursday that the moment was coming where businesses would have to assume there would be no deal.

He said: “If this lack of a solution continues, and approaches the end date, the private sector itself will have to prepare on the assumption that it will be a hard Brexit. That’s where there will be financial uneasiness in markets.”

Cable said he was “not convinced by the Armageddon solution”. He added: “I would play down the possibility of a no-deal and I think there is an element of the government and the Europeans talking this up to sell a bad deal.”

Cable conceded that an extension of the two years of negotiating time allowed under article 50 would probably be necessary to allow a second referendum to take place.

“But if there is a will to make it happen, it will happen. We know that the British government has been preparing because that has been reported, and when I asked Theresa May she declined the opportunity to deny it. I want to hear what the commission is doing.”

Cable said he did not believe attempts by Tory whips to convince Labour MPs to support a deal or else face a no-deal Brexit would work.

“I can’t speak for Jeremy Corbyn but they are not going to prop up the Conservatives and while I know that they are trying to nobble individual Labour MPs, I don’t get the impression they are making much headway,” he said.

“We have a 100 MPs, including 40 Labour MPs and eight Tories, now in support of a people’s vote and it is like an iceberg. There are a lot of others who will be willing to support us as a way out of this mess.”