Theresa May hopes to convince the House of Commons on Tuesday to give her another fortnight’s grace to keep pushing for changes to the Irish backstop – despite the insistence of Michel Barnier that it is Britain that must compromise.

With 45 days to go until Britain is due by law to leave the EU, with or without a deal, the prime minister will address MPs about progress in the Brexit talks, No 10 announced on Monday.

She is unlikely to signal any shift towards a closer future relationship with the EU, after writing to Jeremy Corbyn to underline her continued objections to a customs union, and instead she will focus on the backstop.

“We are absolutely clear on this: we’re not considering Jeremy Corbyn’s customs proposals, we’re not considering any proposals to remain in the customs union. We must have our own, independent trade policy,” May’s spokesman said on Monday.

May will stress her continued focus on the backstop, but the EU’s chief negotiator insisted on Monday there was no question of Brussels giving in to Downing Street’s demands.

“We’re waiting for clarity and movement from the United Kingdom,” Barnier told reporters after talks in Luxembourg with the country’s prime minister, Xavier Bettel.

At a dinner with Barnier in Brussels later on Monday, the UK’s Brexit secretary, Stephen Barclay, had been expected to echo May’s calls for a time limit on the backstop, a unilateral exit mechanism or its replacement with an “alternative arrangement”.

But Barnier suggested beforehand that he would offer Barclay little succour. “Tonight I will repeat the EU’s positions. I will listen to what the secretary of state has to tell us concerning the alternative arrangements which the UK would like,” Barnier said. “But it’s not more than a concept today. I will also evaluate the interest from the UK side for possible changes to the political declaration, which, let me remind you, fixes the outline quite precisely for the future separation.”

Following the dinner, Barnier said there had been “constructive talks” but reiterated the EU’s refusal to renegotiate the Irish backstop. “It’s clear from our side that we’re not going to reopen the withdrawal agreement but we will continue our discussion in the coming days,” he said.

Quick Guide Why extend the Brexit transition period? Show Will the proposal solve anything? The mooted extension to the transition period is a new idea being put forward by the EU to help Theresa May square the circle created by the written agreement last December and the draft withdrawal agreement in March.

That committed the UK and the EU to ensuring there was no divergence between Northern Ireland and the Republic of Ireland.

But it also, after an intervention by the Democratic Unionist party, committed the UK (not the EU) not to have any trading differences between Northern Ireland and Great Britain.

The problem is that these are two irreconcilable agreements. They also impinge on the legally binding Good Friday agreement, which brought peace to Northern Ireland and in some senses pooled sovereignty of Northern Ireland giving people a birthright to be Irish or British or both.

If the UK leaves the EU along with the customs union and the single market then the border in Ireland becomes the only land border between the UK and the EU forcing customs, tax and regulatory controls. The backstop is one of three options agreed by the EU and the UK in December and would only come into play if option A (overall agreement) or option B (a tailor-made solution) cannot be agreed by the end of transition. The Irish have likened it to an insurance policy. The new EU idea is to extend the transition period to allow time to get to option A or B.

But an extension is problematic for Brexiters and leave voters, who want the UK to get out of the EU as soon as possible.

The Irish and the EU will also still need the backstop in the withdrawal agreement, which must be signed before the business of the trade deal can get under way. Otherwise it is a no-deal Brexit.

Extending the transition into 2021 would mean another year of paying into the EU budget. Britain would have to negotiate this but it has been estimated at anywhere between £10bn and £17bn. Staying in the EU for another year would also mean continued freedom of movement and being under the European court of justice, which Brexiters would oppose.

A spokesman for the UK government offered no indication of any progress, adding that the two sides had agreed to further talks and were still seeking to find “a way forward”.

A commission official said the two sides had “discussed the urgent need to find a solution that both respects the EU’s guidelines, as well as one that is supported by a strong majority in the House of Commons. Further engagement is foreseen over the next week.”

Barclay will travel to Strasbourg on Tuesday to set out the UK’s stance to MEPs, and the foreign secretary, Jeremy Hunt, will take the same message to Paris. In Westminster, meanwhile, May is expected to confirm she will hold another Brexit vote in parliament on 27 February.

Opponents of a no-deal Brexit inside her cabinet and on the Tory backbenches want to hear confirmation from the prime minister that any motion she tables then will be amendable. That would allow them to have another attempt at winning a majority for an amendment that could force the prime minister’s hand, if she has not brought a reworked version of her deal back to parliament for MPs’ approval before then.

Yvette Cooper and Nick Boles, two senior backbenchers whose amendment against no deal failed to gain majority support last month, have not yet made a final decision about whether to table a reworked version or wait until 27 February, the Guardian understands. But they have been working on a revised version of their bill to extend article 50, which would then have to be tabled by mid-March to have a chance of being passed before exit day.

Their amendment would set a final mid-March deadline for May to win a majority for her deal, obliging the government to make time for the bill if she has not secured parliament’s support before then.

May’s hopes of cobbling together a majority were boosted on Monday as Boris Johnson said he would be willing to accept a deal with a time-limited backstop. The former foreign secretary told a meeting in parliament: “I think it must be pretty obvious that if you are going to have a time limit to the backstop – and I think that would be very good – it has got to fall before the next election.”

Some more hardline Brexiters have suggested the backstop must be removed altogether from the withdrawal agreement, but that looks highly unlikely to win favour in Brussels. Downing Street hopes that if May can win round Johnson, she can split the hard-Brexit European Research Group (ERG).

Together with a group of Labour backbenchers encouraged by promises of fresh guarantees on workers’ rights and environmental standards, May hopes she can win over enough MPs to pass her deal.

Labour will table its own amendment this week seeking to put what the shadow Brexit secretary, Keir Starmer, calls a “hard stop”, on the government’s negotiating time by enforcing a deadline by which May must bring a deal back to the Commons.

May’s insistence that she will not countenance joining a customs union was seized on by those MPs who want to see Labour embrace a “people’s vote”. They suggested it was now time for Corbyn to do so.

It emerged on Monday that an earlier draft of Corbyn’s letter to the prime minister last week, which was initially drawn up by Starmer, included a reference to a referendum, but it was removed during the editing process.

Asked about the claim by journalists in Dublin on Monday, Starmer said: “The letter was an agreed letter that went out. We’ve now got a response from the prime minister. The critical question is: is she, in her response, indicating a willingness to drop her red lines or not?”

In her letter to Corbyn, May argued that her own Brexit plan “explicitly provides for the benefits of a customs union” in terms of avoiding tariffs, while allowing “development of the UK’s independent trade policy beyond our economic partnership with the EU”.

She wrote: “I am not clear why you believe it would be preferable to seek a say in future EU trade deals rather than the ability to strike our own deals.”