Senior British government sources have said technical discussions around 30 areas of disagreement with the European Union on the Northern Ireland Protocol have resulted in definite progress on just two of the issues.
They say there are "about half a dozen" other areas where some progress towards a solution has been made.
The sources said the British government had not yet given up hope that the EU will move significantly during the next few weeks.
The EU has repeatedly said the UK has failed to deliver on a number of key protocol commitments it signed up to.
The sources said that UK negotiators had submitted ten papers covering a range of proposed solutions they believe would reduce impediments caused by the protocol for businesses in Northern Ireland, but claim the EU has been slow to engage.
The sources say a stock-take of the negotiations will be taken at a meeting of the EU-UK Joint Committee on 9 June.
While new DUP leader Edwin Poots and other unionists insist the protocol must be scrapped entirely, the sources said the British government accepted the need for some mechanism to regulate any new trade agreement on the movement of goods between the EU and Northern Ireland.
"There is going to have to be some form of protocol, if not this protocol," one said.
David Frost, the UK's lead negotiator on the protocol, was in Northern Ireland yesterday to meet business representatives and members of the CNR (Catholic, Nationalist and Republican) community.
He has previously met PUL (Protestant, Unionist and Loyalist) community groups, including the Loyalist Communities Council, which includes representatives of loyalist paramilitary organisations.
Sources said that while the CNR representatives were keen to focus on the opportunities offered by the protocol, rather than the obstacles, there was "empathy" for unionists and loyalists who feel their British identity has been undermined.
"It's clear from the conversations I've had with business and community representatives here today that the way the protocol is operating is presenting very significant challenges," Mr Frost said afterwards.
"I've heard from business groups, as I did last time about how some suppliers from GB are beginning stop sending products into Northern Ireland, we've also heard about reduced availability and choice, cancelled deliveries, products being pulled, interference with the movements of pets, parcels and so on.
"The protocol relies on cross-community support, but this situation does risk undermining that. Our number one priority as the UK government is protection of the Belfast Good Friday Agreement in all its dimensions, so north south and east, west, and indeed that's the explicit top priority in the protocol itself in the text.
"We hope the commission and share that view of prioritisation, though if so they aren't always as clear about it as they could be."
Mr Frost said the major issue that worries the UK is that the EU wants to treat the regulatory boundary in the Irish Sea as if it were like any other external border, despite the obligations of the protocol to facilitate trade between Britain and Northern Ireland.
He said the British government remains committed to working through the issues with the EU to find a way forward "but progress is limited".
"We do think the EU needs to show more common sense and pragmatism, it's hard to see the protocol as currently operating can be sustainable for long," he added.
"We continue to consider all our options in meeting our overriding responsibilities to sustain peace and prosperity in Northern Ireland."
Meanwhile, Tánaiste Leo Varadkar has said that there is "political will in Dublin" to resolve the issues around the Northern Ireland Protocol, but warned that "unilateral action will bring us nowhere".
Speaking at Dublin Castle during an online event in the Shared Island Dialogue Series, Mr Varadkar said that all-island cooperation presents clear advantages which must be used.
He said a solution to problems with the protocol will "require a degree of alignment" on "some rules" and "checks and standards".
He said it is in the interests of the European Commission and the UK "to come to an agreement on this", as the alternative is a "breakdown".
Mr Varadkar said he will do his best to work well with new DUP leader Edwin Poots, however, he said, "political parties are only political parties", and ministers speak for a government and set policy.
He said in order "to reboot" political relationships, there must be agreement on the protocol.
"With the countdown to elections effectively having begun, I think we are in for a turbulent few months - but we have been here before," he added.