A spokesperson for the British Prime Minister says the existing two proposals remain the basis for negotiations.
The government's approach to the Irish border issue is "misguided and fails to recognise numerous central concerns about food", the Chartered Institute of Environmental Health (CIEH) has said.
He added that it was a "legitimate choice" for Britain to be a "friend, a neighbour and a partner" of the European Union, rather than a member.
The UK may have to remain in the customs union in some form, after EU officials rejected the UK's proposal for avoiding a hard border in Ireland.
European leaders are concerned that without a creative solution, there would be a collapse of the Northern Irish peace process, which depends on a soft border between the British province of Northern Ireland and the Republic of Ireland, which will remain an EU member state.
The EU source, who was privy to Wednesday's meeting in Brussels between Britain's lead negotiator Olly Robbins and senior EU officials, reportedly told The Telegraph: "It was made clear that none of the UK's customs options will work".
But government sources did express exasperation at what they regard as the intransigence of the EU27.
Mr Barnier has now suspended internal EU discussions on a future EU-UK trade deal in a bid to force UK concessions before the next European Council meeting in June.
The BBC understands that at a meeting on Wednesday with top United Kingdom officials, the EU's deputy Brexit negotiator, Sabine Weyand, went through the United Kingdom proposals "line by line" and explained why from the European Union side they had not changed sufficiently since previous negotiations and were still unacceptable.
Both sides have committed to avoiding customs checks and other border impediments that might rekindle violence in the British province of Northern Ireland.
Pro soft-Brexit Conservative MPs believe the government may be deliberately presenting proposals it knows will be knocked back by Brussels, to help the prime minister break the deadlock in Cabinet about how to proceed. "The day after, you necessarily have to grasp with reality". I think it was the right thing to do.
The majority of 123 in the customs union vote - double what was expected - was a harbinger of further defeats for the Government in the coming days, as they face a highly organised cross-party opposition in the House of Lords.
Lynn Gilmore - manager of Seafish Northern Ireland, which represents the industry - told the Northern Ireland Affairs Committee at Westminster that tariffs and non-tariff barriers to trade are a major concern.