Britain and the EU failed to make a breakthrough during post-Brexit discussions on trade and the Northern Irish protocol on Wednesday.
The two have been locked in a row in recent months as they accuse each other of failing to honour an agreement late last year on cross-border checks of some goods into the single market.
The argument has been dubbed the "sausage war" because it affects the movement of chilled meats from Britain to Northern Ireland.
In a statement the UK government said there had been no progress on the main issue of the movement of some foodstuffs, adding that "urgent" talks are needed to avoid disruption to critical supplies such as medicines.
"The UK is concerned that substantive progress has not yet been made in many areas," the statement said, listing issues including the movement of food, pet travel and steel quotas.
"We had some pretty frank and honest discussions ... There weren't any breakthroughs. There aren't any breakdowns either, and we are going to carry on talking," Britain's trade negotiator David Frost said.
The EU's chief negotiator warned the UK not to act alone in changes to the Northern Irish protocol and said quotas and tariffs “could come into play” as a response.
"Trust, which should be at the heart of every partnership, needs to be restored," Maros Sefcovic, the vice president of the European Commission overseeing EU-UK relations, said.
"If the UK were to take further unilateral action in the coming weeks, the EU will not be shy in reacting swiftly, firmly and resolutely to ensure that the UK abides by its international law obligations."
"Pacta sunt servanda," he added, using the Latin for "agreements must be kept".
Brussels has accused London of failing to implement agreed checks on some goods moving from Britain to Northern Ireland and has started legal action over London's unilateral extension of a "no-checks" grace period.
London says it has no choice because some of the checks hamper supplies to Northern Irish supermarkets.
It has also pointed to rising tensions among pro-British unionists in the province, who say the protocol undermines the 1998 peace agreement by loosening their ties to Britain.
"We've put forward some solutions to the EU," UK minister Robert Jenrick told Sky News. "It can't be right that ... sausages can't travel from Bangor in Wales to Bangor in Northern Ireland.
Britain is keeping all options on the table in talks with the EU on trade with Northern Ireland, including potentially further extending a grace period on checks on chilled meat moving from the mainland to the province, a senior UK source disclosed to the media.
The source said the British side still believed there were solutions to be found to ease post-Brexit trade with Northern Ireland and the two sides will press on with talks before a June 30 deadline when the grace period ends.
The UK will continue to try and find a solution to the impasse at the G7 summit later this week in Cornwall.