A Sunday Times report speculated that a bilateral treaty between London and Dublin could replace the backstop.
Irish Foreign Minister Simon Coveney tweeted that the Irish Government was committed to the entire withdrawal deal, "including the backstop".
The Guardian reported that government sources said May's "overriding priority was to prevent a historic split in the Tory party".
The divorce deal May had struck the EU was rejected by MPs last week. With just over two months until Britain is due to leave the bloc, some MPs are pushing for the UK to delay its departure until the country's divided politicians can agree on a way forward.
Labour Party Brexit spokesman Keir Starmer said "it's inevitable" Britain will have to ask the EU to extend the two-year countdown to exit that ends on March 29.
"The 29th of March is 68 days away," Starmer told the BBC. "We are absolutely not prepared for it. It would be catastrophic."
Former Brexit secretary Dominic Raab, who quit the government in opposition to May's agreement with the EU, said a no-deal Brexit would have "short-term risks," but they would be "manageable".
British MPs who want a softer Brexit are preparing to try to amend May's plans in a January 29 debate, and to use parliamentary rules to try to prevent a no-deal Brexit and take control of the exit process.
Conservative lawmaker Nicky Morgan said she and several opposition colleagues planned to introduce a bill to ensure "that if the prime minister can't get an agreement approved by the House of Commons by the end of February," the UK will ask the EU to postpone its departure date "so that we can build a consensus and get ourselves more prepared for Brexit. "
Delaying Brexit would require approval from the 27 other EU nations.
Starmer said there was a roadblock in the way of a solution to the Brexit crisis, "and that roadblock is the Prime Minister."
"Her mind is closed," he said.
- AP