The plan emerged after Jo Johnson, a transport minister, resigned from the Government over Brexit and vowed to vote against Mrs May's deal in the House of Commons.
DUP leader Arlene Foster said May's letter "raises alarm bells for those who value the integrity of our precious Union".
The social media post comes as the Democratic Unionist Party (DUP) party responded to May's letter addressed to Foster, which was leaked to the Times.
This backstop - created to avoid a hard border on the island of Ireland - would be the last resort option and come into effect only if future trade negotiations failed to produce a solution for preserving the open Irish border.
This so-called "backstop to the backstop" would see Northern Ireland become wedded to the EU single market and customs union should London and Brussels fail to strike a permanent trade deal.
A Brexit deal on the Northern Irish border could be signed within the "next couple of weeks", the Irish prime minister has said.
The DUP MP Sammy Wilson heaped further pressure on the government, telling the BBC Radio 4's Today programme that the letter was sent to the party but was put into the public domain by Downing Street.
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At the British-Irish Council summit in the Isle of Man, Mr Lidington said: "The prime minister has always been very clear we won't accept something that involves carving out Northern Ireland from the rest of the United Kingdom".
A leaked letter from Ms May to the DUP says the British prime minister wants a "backstop" measure which would create a temporary "joint customs territory" with the European Union for the whole of the UK.
Mr Varadkar said he believed the negotiations were at a "sensitive point", and while a successful outcome is not guaranteed a deal is possible in next few weeks.
The party's Westminster leader, Mr Dodds, warned on Sky News: "I think the prime minister will hopefully realise what can be got through parliament and what can't".
"The time for DUP bluster, chest beating and hand wringing has long gone".
A huge amount of Ireland's trade with continental Europe passed through Welsh ports and across Britain on its way to Calais, while much of Northern Ireland's trade with the British mainland goes via Dublin port, and alternative routes by boat to Rotterdam, Antwerp and French ports were "much slower", he said.
Speaking at the British-Irish Council meeting, Secretary of State Karen Bradley told the Press Association: "The negotiating teams are working hard to get a good deal that can be taken to the British Parliament".
"I think we really have to have regard to that as well".