The UK appeared to inch closer to a deal over the post-Brexit Northern Ireland Protocol after “productive” talks were held during an “intensive” phase of negotiations.

Foreign Secretary James Cleverly and the EU’s Maros Sefcovic agreed to hold a face-to-face meeting in the coming days after a video discussion on Monday.

Downing Street earlier insisted a “final deal” had not been struck as they seek an agreement that will satisfy the demands of Northern Ireland’s Democratic Unionist Party (DUP).

No 10 would not commit to giving MPs a vote on any potential deal as senior Conservative Brexiteers stepped up warnings to Rishi Sunak not to give too much ground to the EU.

Sir Keir Starmer urged the Prime Minister to allow the Commons to have its say as he reaffirmed his offer of Labour support to secure the approval of any new agreement amid signs there could be a rebellion from some wings of the Tory party.

Mr Cleverly welcomed Monday’s discussions with the European Commission vice-president and tweeted: “Intensive work continues and we agreed to talk again in the coming days.”

Mr Sefcovic described their video call, which was also joined by Northern Ireland Secretary Chris Heaton-Harris, as a “productive” attempt to find joint solutions.

“Hard work continues. We’ve agreed to meet later this week,” he tweeted.

Sources in Brussels welcomed the move to schedule in-person talks as a positive step, but said a location had not been set.

Earlier in the day, Conservative former minister Simon Clarke backed a call by Boris Johnson for ministers to press on with legislation enabling them to override parts of the protocol.

But the Prime Minister’s official spokesman stressed the Northern Ireland Protocol Bill, which is currently stalled in the Lords, remains “important” in “the absence of a negotiated solution”.

The spokesman told reporters: “It’s clear that we need to find solutions that protect Northern Ireland’s place in our internal market, safeguard the Good Friday Agreement and resolves the practical issues that the protocol is causing for families and businesses.

“The Prime Minister has been clear that we have not resolved all of those issues and no deal has been done as yet.”

Meanwhile, Home Secretary Suella Braverman described the Protocol Bill as “one of the biggest tools that we have in solving the problem on the Irish Sea”.

Ms Braverman, a longstanding Eurosceptic, argued that Mr Sunak is right to be “committed to finding a pragmatic solution to resolve these issues”.

The Prime Minister’s spokesman said Downing Street will remain “in close contact” with the DUP and other parties, but added: “It would be wrong to say there is a final deal.

“There’s intensive work to do with the EU, that work is ongoing.”

Downing Street declined to commit to a Commons vote, with the official spokesman only saying: “You will hear more from us should an agreement be reached with the EU that reaches the challenges the Prime Minister has set out.”

Sir Keir, speaking to broadcasters during a visit to Thurrock in Essex, said: “I think it should be put to a vote, and I’m very clear to the Prime Minister I will put the country first and the party second and he should do the same.”

Veteran Tory Eurosceptic Sir Bernard Jenkin said that any deal which did not lead to a return to powersharing at the Stormont Assembly by the DUP – which walked out in protest at the protocol – would be “completely disastrous”.

“If it doesn’t get the support of both communities in Northern Ireland it is just going to make things worse because it will cement in place an agreement that has destroyed powersharing in Northern Ireland,” he told Times Radio.

Sammy Wilson, the DUP chief whip at Westminster, accused ministers of going into the negotiations with the EU with “an attitude of defeat” and said that he did not expect to see a deal announced this week.

“If a deal is agreed which still keeps us in the EU single market, as ministers in the Northern Ireland Assembly we would be required by law to implement that deal and we are not going to do that because we believe such an arrangement is designed to take us out of the United Kingdom,” he told Sky News.

“We are British and we expect to be governed by British law, not Brussels law. We would certainly not collaborate in administering Brussels law in our part of the United Kingdom.”

The more measured response of DUP leader Sir Jeffrey Donaldson was privately welcomed in Downing Street.

It is absolutely imperative tactically to give our negotiators the strongest possible hand to play with Brussels

Mr Clarke, the former levelling-up secretary, argued the Protocol Bill would strengthen the hand of ministers in negotiations and could ultimately provide a better solution than a new deal with the EU.

“It is absolutely imperative tactically to give our negotiators the strongest possible hand to play with Brussels,” he told BBC Radio 4’s Today programme.

“If the perception is there that the Bill is moribund then that will, I am afraid, weaken our hand very considerably.

“We need to make sure that if a deal is struck here it is genuinely a better one than that we can achieve through our own legislation to fix the protocol.”

The Conservative Eurosceptics in the European Research Group (ERG), who could provide resistance to Mr Sunak’s attempts to pass a new deal, will meet for talks on Tuesday.

Mr Johnson’s intervention over the weekend – his first on Brexit since departing No 10 – has raised concerns over a potential rebellion by Eurosceptic Tory backbenchers if Mr Sunak’s changes are put to a vote in Parliament.

Some Tories quickly sided with the former prime minister, with Lord Frost – who negotiated Mr Johnson’s original Brexit deal – urging the Government to “push on with the Protocol Bill”.

However Mr Clarke, a strong supporter of both Mr Johnson and his successor Liz Truss, denied he was trying to make trouble for Mr Sunak.

“Quite the opposite. I want to see this issue fixed. It is vital for the Conservative Party,” he said.