Anti-Brexit demonstrators wave EU and Union flags outside the Houses of Parliament in London, Britain, January 30, 2018.
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.
Failure to avoid a hard border would likely see the European Union refuse to sign a withdrawal agreement and leave Britain out of the bloc next March without even so much as a transition period.
The Irish government has warned that any Brexit deal will be torpedoed if the issue of the Irish border is not resolved.
When it comes to the UK's future trade relationship with the EU, Theresa May has already said that she hopes to secure a deal that's good for both sides.
Mrs May's Brexit "war cabinet" convened again on Tuesday, but did not reach an agreement on which of the two options for customs arrangements on the Irish border - the "customs partnership" or "maximum facilitation" - it will support.
The UK authorities will inform Brussels about the willingness to remain in the customs Union of the EU after 2021 due to the unresolved Irish question.
In a sign of her determination to inject fresh impetus into the process, May brought chief Brexit negotiator Olly Robbins, chief of staff Gavin Barwell and her principal private secretary Peter Hill to her meeting in Sofia with Tusk and European Commission President Jean-Claude Juncker.
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Mr Varadkar expressed cautious optimism the proposals could be acceptable to the Irish government.
Varadkar said: "We need to have that backstop because that gives us the assurance that there will be no hard border on our island".
Ministers including Boris Johnson, Michael Gove and David Lidington discussed the "backstop" plan for Northern Ireland that will come into force if Mrs May's blueprint for customs arrangements collapses. It would keep the United Kingdom in the EU's customs arrangement until such a time as another solution was found. Leo Varadkar said he would not disclose exactly what Ms May had offered, but made it clear that the plan from the British was insufficient.
However, Ireland's Leo Varadkar told May that Britain must also stay aligned with the single market for there to be no physical infrastructure on the border between Northern Ireland and the Republic.
They also opinioned that for things to settle in properly the pending issues like the quandary at the Irish border have to be cleared on an urgent basis.
The Government's aim is to publish the white paper in time for the next EU summit of European leaders in six weeks' time.