Many in Northern Ireland and the Republic of Ireland are anxious that an indefinite backstop could unpick the carefully crafted provisions for North-South cooperation set out in the 1998 Belfast Agreement.
Ms May said opposition leader Jeremy Corbyn "also believes the potential indefinite nature of the backstop is an issue", and that the European Union has "already accepted the principle of "alternative arrangements" superseding the backstop should it ever be required".
UK MPs last week backed a call for May to return to Brussels to secure a new Brexit deal that puts "alternative arrangements" in place of the controversial backstop - a kind of insurance policy created to avoid a hard border between Northern Ireland and the Irish Republic.
According to Lord Trimble, such arrangements would include replacing the backstop with an interim free trade agreement in goods and agri-food, with a "chapter on Customs and Trade Facilitation, and Irish Border Facilitations".
The Brexiteer would also introduce new legal text on a raft of areas such as zero tariffs in goods and agriculture, and regulatory coherence covering specific sectors, such as the pharmaceutical industry.
In what will be the prime minister's third visit to Northern Ireland in seven months, Mrs May will this afternoon make a speech about Brexit at a critical moment in the process of the UK's leaving the EU.
"These emergency evacuation plans have been in existence since the Cold War, but have now been repurposed in the event of civil disorder following a no-deal Brexit", Britain's The Sunday Times said, quoting an unnamed source from the government's Cabinet Office, which handles sensitive administrative issues.
In her speech, Mrs May will say she will find a solution "that commands broad support across the community in Northern Ireland" and "secures a majority in the Westminster Parliament, which is the best way to deliver for the people of Northern Ireland". "It is in all our interests to get to that agreement and for the European Union to say we are not going to even discuss it seems to me to be quite irresponsible".
Britain on Monday will seek to reassure foreign investors anxious about Britain leaving the European Union (EU) after Japanese automaker Nissan said it was scrapping a major planned investment less than two months before Brexit.
Critics fear the United Kingdom could be "trapped" in this arrangement for years, while Unionist MPs in Northern Ireland also fear it would divide Northern Ireland from the rest of the UK.
The first meeting on Monday was described as "detailed and constructive" by the Brexit department.
Mrs Foster said Parliament had spoken and the current backstop has been rejected.
"As we look toward a new partnership with Europe, it is our shared values and commitment to each other, that are out greatest asset, and demonstrate that even through change, our enduring alliance remains strong, and as innovators, traders and internationalists we look with confidence to the future".
Senior European Union figures have strongly rejected calls for the Withdrawal Agreement to be rewritten to remove the backstop.
Irish deputy premier Simon Coveney said alternatives to the backstop were "wishful thinking". "But we are ready, during the transition, if we have a transition, to work on the alternative arrangements", Barnier said.
He said: "The Irish protocol in the Withdrawal Agreement already allows for alternative arrangements or alternative solutions to the backstop and if they're there they can replace the backstop".
German chancellor Angela Merkel struck a more emollient tone.