British ministers, The Sun newspaper said, have been examining a plan drawn up by Japan's Fujitsu to track trade across the border, while the Telegraph said ministers had discussed delaying Brexit by eight weeks.
Albert Einstein is sometimes credited with having said that the definition of insanity is doing the same thing over and over again and expecting different results.
But May's spokesman, James Slack, denied that the Brexit process was deadlocked.
Still, May plans to meet with European Commission President Jean-Claude Juncker and other EU leaders in Brussels on Thursday seeking changes to the backstop, and will return to Parliament next week with - she hopes - a modified plan.
Speaking to business leaders on Tuesday during a two-day trip to the region, May said she understood local concerns about the current deadlock in Westminster, where Conservative MPs continued to discuss "alternative arrangements" to the backstop - the backup mechanism created to avoid a hard border set out in the withdrawal agreement May struck with the European Union (EU).
May said she would seek an alternative arrangement which avoids the need for a hard border or legally binding changes to the border backstop to introduce a time limit or create an exit mechanism.
Border disruption caused by a No Deal Brexit would also pose significant risks to the museum sector, the statement said. It would also involve a temporary single customs territory effectively keeping the whole of the United Kingdom in the customs union.
The comments came as Justice Secretary David Gauke became the latest Cabinet minister to suggest that Brexit might have to be delayed beyond the scheduled date of March 29, telling the BBC that the "key" was achieving a "smooth and orderly departure".
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May said there will be no customs border between Great Britain and Northern Ireland, adding that she will strengthen bilateral relations between the United Kingdom (UK) and the Republic of Ireland.
"We expect that the backstop would never be used", Varadkar said.
The football associations of England, Scotland, Wales, Northern Ireland and Ireland are still in talks over whether to proceed with a bid, with a year-long feasibility study needed before any decision is taken.
The company said it had made the decision "for business reasons", but added that "the continued uncertainty around the UK's future relationship with the European Union is not helping companies like ours to plan for the future".
"The UK needs to be sure that foreign products can be imported and UK products exported as usual", the government said in a statement to parliament.
Without question, digital technology can play a growing role in border tracking and management. Several European populist parties have backtracked on this issue.
A third challenge for May is the significant demands of her Conservative and DUP critics.
Fourthly, this issue is also politically charged in Ireland beyond the DUP.
Ms DeSouza holds an Irish passport and claimed the requirement undermined her right to be an Irish citizen under the Good Friday Agreement. He, however, also warned that no-deal preparations were taking place in collaboration with the Irish PM in case of a negotiation "fiasco".
Speaking in Northern Ireland on Tuesday, May promised to seek a solution that keeps the Irish border open and preserves the peace agreement that ended years of sectarian violence in the British province.