Theresa May reiterated her opposition to a second Brexit referendum on Monday night, claiming it would threaten Britain’s “social cohesion” and insisting the centrepiece of her strategy remained negotiating changes to the Irish backstop.

With just 67 days to go until Britain is due by law to leave the European Union, May exasperated MPs and business groups by offering scant evidence that she was willing to change course.

Giving a statement in the House of Commons, the prime minister outlined three changes she claimed had emerged from discussions with colleagues in the six days since her Brexit deal was rejected by MPs with a crushing margin of 230:

• A more consultative approach to the next phase of negotiations, with MPs, business groups and unions more involved.

• Stronger reassurances on workers’ rights and environmental standards, “with a guarantee that not only will we not erode protections for workers’ rights and the environment but we will ensure this country leads the way”.

• Another attempt to address the concerns of Tory and Democratic Unionist party MPs about the Irish backstop – which she could then discuss with Brussels.

May dismissed the idea of extending article 50 and stepped up warnings about the potential consequences of asking the public to vote again on Brexit.

“There has not yet been enough recognition of the way that a second referendum could damage social cohesion by undermining faith in our democracy,” she said.

Theresa May at 10 Downing Street, London. Photograph: Mark Thomas/REX/Shutterstock

Afterwards, May’s spokesman said: “There is a covenant of trust between the electorate and the government of the day and the PM’s firm belief is that it is the government’s duty to act on clearly expressed wishes of the electorate and, obviously, were that not to happen, that wouldn’t be, and shouldn’t be, without consequence.”

May flatly rejected the idea of ruling out a no-deal Brexit, claiming the only way to do so was to accept her deal – or revoke article 50 altogether.

But the prime minister faces a looming revolt over the issue, with cabinet ministers and other Tory frontbenchers likely to step up calls for a free vote on an amendment put forward on Monday night by Labour’s Yvette Cooper that could pave the way for an extension of article 50 if no agreed deal has been reached.

The work and pensions secretary, Amber Rudd, is understood to have warned May that resignations from the front bench may follow, if the prime minister did not allow ministers to express their backing for the move.

Tobias Ellwood, a defence minister, also tweeted his tacit support. “Cooked a banana cake yesterday. Told my son it will be ready in 20 mins – according to the cookbook. It took 30. It was a big decision – honouring the cookbook or take more time to get the right result,” he wrote.

Business groups reacted with alarm to the lack of new thinking in May’s statement, with the CBI director general, Carolyn Fairbairn, calling it “another bleak day for business”.

“The government’s move to consult more widely is welcome, as is the commitment to scrap the settled status charge for EU citizens, but the fundamentals have not changed. Parliament remains in deadlock while the slope to a cliff edge steepens,” she said.

May claimed she had met MPs and other parties “in a constructive spirit, without preconditions”, and criticised Jeremy Corbyn for refusing to take part unless she ruled out a no-deal departure.

But the Labour leader said her talks had been a “PR sham” and accused her of being in “deep denial” about the scale of defeat in last week’s meaningful vote.

“The logic of that decisive defeat is that the prime minister must change her red lines because her current deal is undeliverable. So can she be clear and explicit to the house: which of her red lines is she prepared to move on?” Corbyn asked.

Labour later tabled its own amendment to May’s Brexit motion, calling for the government to put in place a process for choosing between possible options – including Corbyn’s own policy – and a “public vote”.

Hilary Benn, the Labour chair of the Brexit select committee, tabled his own amendment which would allow indicative votes on four options set out by the committee, May’s deal, no deal, a renegotiation based on a Canada or Norway model, and a second referendum.

It was unclear what changes to the backstop May hopes to secure, but the Tory backbencher Andrew Murrison is understood to be in advanced discussions about tabling an amendment time-limiting the backstop, after hearing May tell the Commons that “the length of the backstop was being actively considered”.

Should he decide to table an amendment, it may call for a five-year limit as suggested by the Polish foreign minister earlier on Monday. Murrison had attempted to table an amendment to the prime minister’s Brexit deal last week, putting a time-limit on the backstop, but it was not selected by the Speaker.

The prime minister also announced on Monday that the controversial fee for EU nationals to register to stay in Britain after Brexit will be waived, after a backlash from citizens’ rights groups and MPs from across the spectrum of Brexit opinion.

Under the planned scheme for EU nationals to apply to stay in the UK, which is currently being piloted, those aged over 16 have to pay £65, with a cost of £32.50 for anyone younger.

Cooper, the chair of the home affairs select committee, put down a tightly-worded amendment on Monday night to give time for a bill that would give parliament the power to support an extension of article 50.

A more radical amendment is planned by the former attorney general Dominic Grieve, which would allow any motion put forward by a minority of 300 MPs from at least five parties – including 10 Tory MPs – to be debated in the Commons the following day.

It is understood that some MPs present expressed concern that Grieve’s amendment may not attract the support of the Labour frontbench, who are concerned about its major constitutional implications. “We really need Labour to whip for this for it to have any chance of winning,” one source present said.

Second referendum campaigners are expected to focus their efforts on the moves to extend article 50, rather than tabling their own amendment. “A people’s vote will probably not secure a majority in the House of Commons until every Brexit option has been exhausted, but there will be multiple opportunities in parliament to give the public the final say when it has become clear this is the only way forward,” a spokesman said.