Cabinet ministers are bitterly divided over whether Brexit talks with Labour should broach the possibility of a customs union, with several sceptical that such a deal could even command a majority in parliament or survive hostile backbench amendments.

A senior cabinet minister suggested a deal involving a customs union could be backed by as few as 90 Tory MPs and would mean a slew of resignations from the government payroll.

It is also likely to be opposed by the SNP, the Liberal Democrats and other smaller parties, as well as dozens of Labour MPs who would only back a deal if it included a confirmatory referendum.

Concerns that a deal with a customs union would still not command a majority were publicly aired by the foreign secretary, Jeremy Hunt, on Tuesday, one of the cabinet member’s who is most strongly opposed to a customs union. Other opponents include the international trade secretary, Liam Fox, and the Commons leader, Andrea Leadsom.

Other ministers are more optimistic, with one cabinet source saying they believed up to 160 Tory MPs would eventually back a customs union deal.

“You get 60 just from the One Nation group – they would back a deal with a customs union,” the source said, referring to the group of soft Tories backed by ministers including David Gauke, Amber Rudd and Claire Perry.

At a cabinet meeting on Tuesday both the environment secretary, Michael Gove, and the chief whip, Julian Smith, were said to have urged ministers to accept that a deal with Labour may be the only path to delivering Brexit. “Julian clearly thinks the numbers are there,” one cabinet source said.

However, there were also grave concerns that no alliance with Labour would be solid enough to hold to pass the legislation required – or to withstand any hostile amendments being successfully added to the bill.

“If I were [the Labour chief whip] Nick Brown, I would say that after it was amended, the deal has changed and it’s no longer the deal we agreed to,” one cabinet minister said.

Theresa May told the liaison committee on Wednesday that she had an “open mind” on a deal to be done with Labour. “There are differences on issues but on many of the key areas – particularly on the withdrawal agreement – there is common ground,” she said.

The prime minister added: “We know that we need to end this uncertainty and do it as soon as possible and I hope a deal can be done. We certainly approach this with an open mind. But if we are not able to do that, then we will bring votes to the house in order to determine what the house will support. We stand ready to abide by that decision if the opposition are willing to do so.”

Jeremy Corbyn’s spokesman was positive about the talks, describing them as “serious and constructive” and expressing optimism they could reach a conclusion.

He told reporters: “I think we’ve seen clear evidence that the government is prepared to explore shifts in its position. And that’s what we need to nail down – how far and where those shifts are going to go. Any idea that they’ve reached some kind of breakdown or deadlock is not correct.”

There will be more official-level and one-to-one ministerial meetings this week, with a meeting of the full delegations likely to take place next Tuesday that Downing Street has hinted is likely to be key if a deal is to be agreed.

“From the basis of the point they’ve got to this week, I would say that there is significant movement being discussed,” Corbyn’s spokesman said, though he suggested it would be unlikely anything could be sealed in time to prevent European elections on 23 May.

It is understood that Gove’s role has been crucial, acknowledging the need to find ways of locking in any guarantees, so that they could not be immediately unpicked by a future Tory leader.

Gove is said to have told cabinet ministers at their meeting on Tuesday that the risks of losing Brexit entirely outweighed an “unpalatable” customs union.

Both sides acknowledged that there was now a brief window to strike a deal and expected the pace to pick up once this week’s local elections were out of the way.

They have also discussed how backbench-led amendments could be used as part of the process of offering reassurances to Labour.

The shadow business secretary, Rebecca Long-Bailey, who is part of the Labour negotiating team, indicated a solution could be found. “I think pragmatically that they potentially may have no option in order to be able to push this deal through,” she told the BBC Radio 4 Today programme.

She suggested Labour wanted to see movement from the government first before considering whether it could also compromise. “We are fleshing out the details to see how far the government can move towards us and then we will be able to ascertain how far we are able to move towards them,” said Long-Bailey.

MPs have speculated that Labour could agree a deal with the government but then whip the party to back an amendment by Peter Kyle and Phil Wilson to add a confirmatory referendum to the deal but with the expectation that it would fail without significantly more Tory backing.

The former Labour cabinet minister Ben Bradshaw, a supporter of the people’s vote campaign for a second referendum, said MPs’ views on a fresh poll were now so entrenched that few would be prepared to back any deal that did not trigger a public vote.

He said: “There is no path to a deal, including one with a customs union, without a confirmatory referendum. It would be opposed by most Labour MPs. If anything, views have hardened in recent weeks, especially given May’s successor will not feel bound by any deal struck now.

“It would also split both parties, lead to resignations of shadow frontbenchers and to a further loss of Labour members and votes to the smaller parties that back a public vote. I cannot conceive that Jeremy [Corbyn] would want to risk all that and, along with it, the possibility of a Labour government. The only way out of this mess will be public vote.”