The Trades Union Congress is prepared to campaign for a referendum on a Brexit deal unless Theresa May returns with an agreement that is good for British workers, the organisation’s general secretary has said.

Frances O’Grady said that unless the prime minister secured a deal that protects jobs and workers’ rights, the umbrella organisation for the UK’s unions would support the People’s Vote campaign.

Her comments came as union delegates gathered in Manchester for the annual TUC conference, where Brexit will be debated on Monday, and two weeks before the Labour party conference in Liverpool, where delegates are expected to debate and vote on Brexit policy.

Although her proposal has been opposed by some union leaders, it will add to the pressure on Jeremy Corbyn to back a referendum on the final Brexit deal if a general election is not called.

O’Grady told the BBC’s Andrew Marr Show: “I want to serve notice to the prime minister today that if we don’t get the deal that working people need, the TUC will be throwing our full weight behind the campaign for a popular vote so that people get a say on whether that deal is good enough.

“People don’t trust politicians. They are really worried about what Brexit means for their jobs, their communities and their wage packets with prices going up too. If the prime minister can’t get a deal that meets our needs then we will be campaigning for a popular vote.”

O’Grady called on May to seek an immediate extension on article 50 so she could attempt to negotiate a deal that would protect the rights and jobs of UK workers.

“Time is running out and crashing out of the EU would be an absolute disaster for the people we represent,” she said.

In a further boost for the People’s Vote campaign, the public service union Unison confirmed that it would also support a second referendum under certain circumstances.

A spokesperson for Unison, which has 1.3 million members, said: “When and if an agreement is reached, asking the public for their views is definitely one option, as is thorough parliamentary scrutiny. But the best option of all would be a general election.”

It means the three biggest unions – Unison, GMB and Unite – have all indicated their willingness to support a second vote on Brexit in the absence of a general election.

One union leader who opposes a second referendum said on Sunday that it could spark widespread civil disobedience. Appearing at a TUC fringe event, the RMT general secretary, Mick Cash, said trade unionists had voted for Brexit in droves and should not be ignored.

“The only vote now that matters is a general election and the sooner the better. The talk of a second vote on the withdrawal terms is a de facto second referendum whichever way you try to dress it up and risks becoming a Trojan horse for civil unrest on our streets,” he said.

A poll in the Observer disclosed that members of Britain’s three biggest trade unions now support a new referendum by a margin of more than two to one.

The survey of more than 2,700 members of Unite, Unison and the GMB by YouGov for the People’s Vote campaign also found that a clear majority of members now backed staying in the EU, believing Brexit would be bad for jobs and living standards.

In an interview with the Observer before the poll findings were released, the shadow chancellor, John McDonnell, said his preferred option was still for voters to be offered a say on the government’s handling of Brexit – and any deal brought back from Brussels by May – in a general election. But he said that if Labour was unable to force one in the coming months, he wanted to “keep all options open”.

In comments that will increase the pressure on May to seek a deal from the Brexit negotiations, the CBI’s director general, Carolyn Fairbairn, said the lack of progress over the summer had left business leaders frustrated and “bordering on despair”.

“When people went away for the summer they hoped that we would come back for the new term in September with a renewed sense of purpose and instead we’ve come back to siren voices, you know, the soap opera.

“If you’re running a business at the moment you can only plan for one outcome and that is a no-deal scenario and that is what we’re seeing,” she told the BBC’s Pienaar’s Politics.

She urged the government and the EU to take the possibility of a no-deal Brexit off the table or face “catastrophe”.

“There are 150,000 businesses out there that have only ever traded with the European Union, they have no systems in place, they have no ability to be able to deal with WTO rules, they have no ability to be able to get their stuff across borders.

“There are many businesses who don’t know if their goods would be legally tradable in a no-deal world. This is an incredibly serious thing to be considering as a possibility,” she said.