The MPs want to replace the current backstop proposal with a technology-based solution for managing the flow of people and goods across an invisible Irish border.
In a letter to her party's lawmakers, May said she's planning to meet with European Commission President Jean-Claude Juncker and speak to the leader of every EU member state in the days ahead.
Commissioner for Economic and Financial Affairs, Taxation and Customs Pierre Moscovici said: "With the risk of a no-deal Brexit increasing as we get closer to March 29, the European Commission and national customs authorities are working hard to be ready to introduce checks and controls on goods flowing between the EU and the UK".
The changes sought by Mrs May is convince MP's to vote through a Brexit deal in the House of Commons.
Pro-Brexit Tories are demanding that the backstop be removed from the Withdrawal Agreement, but in a move that will likely inflame tensions on the government benches culture secretary Jeremy Wright yesterday suggested that any reform may well stop short of scrapping the backstop.
Meanwhile, campaigners for a second referendum on Brexit have said there will be a major protest the weekend before Britain's scheduled departure date.
'I do not underestimate how deeply or how sincerely colleagues hold the views which they do on this important issue, ' she wrote.
The European Union said it would be "happy" to offer clarifications on the Brexit deal that might help build a majority for the accord in the British Parliament while reiterating that renegotiation is off the table.
The strong warning from Make UK, previously known as the EEF, comes as Japanese carmaker Honda is expected to say it is preparing to shut its main UK plant with a loss of 3,500 jobs.
'If this is the only way of doing it then that's the way we will pursue.
"I don't think it's the mechanism that matters, it's the objective", he told the BBC's Andrew Marr Show, when asked whether a codicil - a supplementary document explaining or modifying a legal agreement - might work.
'That's what Parliament has been very clear that it wants, it will back this deal if we can do something about the backstop'.