Theresa May has insisted the UK will leave the EU customs union amid speculation the departure could be delayed for years until a replacement system is agreed.
The Cabinet remains divided over a solution to the issue and neither of the two proposed models being considered by ministers is acceptable to Brussels.
The Brexit "war cabinet" met on Tuesday without reaching agreement on which of the two options for customs arrangements - the "customs partnership" and "maximum facilitation" models - it will back.
The Daily Telegraph reported that Britain will tell the European Union it is prepared to stay tied to the customs union beyond 2021 as a last resort to avoid a hard border between Northern Ireland and Ireland.
Number 10 sources insisted "there was not a proposal discussed or agreed that would see us staying in the customs union beyond the implementation period", which is due to expire at the end of 2020.
Arriving at the EU Western Balkans summit in the Bulgarian capital Sofia, Mrs May denied climbing down over membership of the customs union.
She said: "No we are not (climbing down). The United Kingdom will be leaving the customs union, we are leaving the European Union.
"Of course we will be negotiating future customs arrangements with the European Union and I have set three objectives, the Government has three objectives in those.
Eurosceptics responded with dismay to the suggestion the UK could remain closely aligned to Brussels.
Jacob Rees-Mogg, leader of the influential European Research Group, told the Telegraph: "The risk of the Government using all its mental energy on the fallback position is that they create a position that is more attractive than a permanent deal.
"We have gone from a clear end point, to an extension, to a proposed further extension with no end point."
But former cabinet minister Damian Green, who backed Remain in the referendum, said he was ready to accept a "small delay" to Brexit to ensure customs arrangements work effectively.
The Ashford MP said:
Former minister Nick Boles said the reported proposal "sounds like a sensible and pragmatic way forward" while Leave-supporting Geoffrey Cox said "pragmatic compromise will be necessary".
The next crunch point in negotiations with Brussels is rapidly approaching and Mrs May was using her visit to Sofia to hold talks with key players including German Chancellor Angela Merkel, French President Emmanuel Macron, European Council president Donald Tusk and European Commission chief Jean-Claude Juncker.
Irish premier Leo Varadkar, who was meeting Mrs May at the Sofia summit, warned that the EU Withdrawal Agreement could be in jeopardy unless substantial progress is made by the June 28 gathering of the bloc's leaders.
Arriving at the Sofia summit the Taoiseach said: "If we are not making real and substantial progress by June then we need to seriously question whether we're going to have a Withdrawal Agreement at all."
He added that it was an "absolute red line" for Ireland not to have a hard border on the island and all 27 remaining EU states supported the current "backstop" proposal in the Withdrawal Agreement - which Mrs May has said no British prime minister could accept.
"If the UK wants to put forward alternatives to that - whether it's an alternative text to the backstop or whether it's some some sort of alternative relationship between the UK and the EU - we are willing to examine that.
"But we need to see it written down in black and white, we need to know that it's workable and legally operable and we have yet to see anything that remotely approaches that."