Theresa May has offered a sop to rebellious hard Brexiters by suggesting technological solutions previously dismissed by No 10 could ultimately be used to maintain an invisible border between the UK and Ireland.

The unexpected change arose on Tuesday during a two-and-a-half hour cabinet meeting at which, the prime minister’s spokesman said, ministers had discussed “the potential for alternative arrangements to avoid a hard border” in Ireland.

The spokesman said one of the alternative arrangements “could involve technological solutions” – a suggestion reminiscent of the “maximum facilitation” model pushed by hard Brexiters but ditched by No 10 earlier this year.

May’s decision to put it back on the table comes as she tries desperately to prevent rebellious Tory MPs from sending in enough letters to the party’s backbench 1922 Committee to trigger a confidence vote.

The number of letters publicly submitted to the committee chairman, Graham Brady, declaring no confidence in May had stalled at 26 on Monday, well short of the 48 needed, though Eurosceptic MPs have insisted more have been submitted privately.

Earlier on Tuesday, Jacob Rees-Mogg called on Tory backbenchers to push for a vote of no confidence immediately, otherwise May would remain in post for years. “I think it is now or the prime minister will lead the Conservatives into the next election,” he said.

Play Video 0:53 Jacob Rees-Mogg on vote of no-confidence: 'Patience is a virtue' – video

May is due to head to Brussels on Wednesday afternoon for further negotiations with Jean-Claude Juncker, the European commission president, on the final political declaration for the future relationship between the UK and the EU.

No 10 would not say when the declaration would be published, although both it and the already published legally binding withdrawal agreement could be signed off in Brussels at a special European summit on Sunday.

A pamphlet published by the thinktank Global Britain and the European Research Group restated the case for “max-fac” on Tuesday morning, a model under which customs checks enabled by technology happen away from borders.

No 10 had previously rejected it after tax chiefs warned it could cost up to £20bn a year to implement. There was also widespread scepticism outside pro-Brexit circles that the technology was ready.

But with May under intense political pressure, No 10 said it was “looking at what alternative arrangements might exist” and hoped one of them could eliminate the need for the Northern Ireland backstop.

The UK and the EU hope to reach a long-term, free-trade deal to avoid a hard border returning between Northern Ireland and Ireland. The previously stated alternatives to that involve extending the transition period beyond 2020, or deploying the backstop, which is unpopular on the Tory right and with the Democratic Unionist party in Northern Ireland, which props up May’s government.

At a morning lobby briefing for journalists, the spokesman said: “I think there was discussion in cabinet about the fact that the withdrawal agreement recognises and keeps open the potential for alternative arrangements to avoid a hard border on the island of Ireland.

“Both the text of the Northern Ireland protocol itself and the outline political declaration note, and I quote: ‘Note the [European] Union’s and the UK’s intention to replace the backstop solution on Northern Ireland by a subsequent agreement that establishes alternative arrangements for ensuring the absence of a hard border on the island of Ireland on a permanent footing’.”

EU officials have insisted that the price of having an all-UK customs union as a backstop was that it would be a “bridge” to the future relationship, as laid out in the jointly published outline of the political declaration, as this was the only legal way to include such an arrangement in the withdrawal agreement.

One official said: “In negotiations on the future relationship [it says] we are going to build on the single customs territory ... and that really speaks for itself. It goes without saying. The EU has had difficulty in establishing a customs union in the withdrawal agreement and we think this is pushing article 50 to the limits. What we can do on article 50 is you can have a phasing out process in terms of orderly withdrawal, or you can also make a bridge to the future, but you have to know where that bridge is leading.

“The condition for having the single customs territory is, we said, that is the starting point for the future customs negotiations.”