The Times reported in Monday's edition that May favors a close partnership deal in which the United Kingdom conducts the EU's customs checks on the bloc's behalf, but that some of her ministers believe this won't work.
"The Elders welcome the assertion by UK Prime Minister Theresa May of the paramountcy of the Good Friday Agreement, and the importance of ensuring that nothing should be done to allow the reimposition of a hard border".
May's inner circle thinks she could be forced to accept staying in the EU's customs union because Parliament will reject her plan to withdraw from it when the issue comes to a vote in the House of Commons, according to one official.
He said: "People say there is no solution to Ireland - I say there is a solution to Ireland because there is a back stop".
The EU's chief Brexit negotiator emphasised that for the EU27, only 6% of trade is with the United Kingdom, while 60% of trade takes place within the single market; this means that the economic impact of Brexit will be limited he argued. The UK will be liable to pay a settlement of around 39 billion pounds to the European Union upon its withdrawal from the bloc.
Not only was an insufficient onus placed on Brexit campaigners to specify the country's future trading terms in great precision, including Northern Ireland's future border arrangements, but the actual ballot paper only asked voters: "Should the United Kingdom remain a member of the European Union or leave the European Union?"
Last week's defeat, the first of two on that day, increased pressure on May to reconsider her refusal to remain in a customs union with the EU.
Stephen Kelly, the chief executive of Manufacturing NI, said it would be impossible to prevent the products - which are banned by the European Union - from slipping into the Republic of Ireland, which made customs checks inevitable.
"These sensible limits do nothing more than hold the Government to their word about continuity of rights after Brexit".
Following the hearing, Labour MP Pat McFadden, who sits on the committee and supports campaign group Open Britain, said: "[Davis] refused, several times, to repeat his previous threat that this was a take it or leave it vote and that rejection of the overnment's resolution would mean "the deal falls.
European Union leaders are due to assess progress on the border issue when they meet at the end of June. A vote in the chamber could come as soon as next month when an amendment on the Trade Bill is debated and enough Tories have signed up to it to defeat May. "The prime minister is absolutely clear: we are leaving the customs union and will be free to strike our own trade deals around the world", he said.
The government says the charter's rights are already covered by British law.