The government will come under pressure on Monday to give a legal guarantee that there will be no border down the Irish Sea after Brexit.
The move is supported by the Democratic Unionist Party (DUP) in an amendment to the Customs Bill which is due before Westminster.
The party is also due to meet the government this week about the prime minister’s Brexit plan.
On Friday, the government published its blueprint for UK relations with the EU.
The long-awaited White Paper is aimed at ensuring trade co-operation, with no hard border for Northern Ireland, and global trade deals for the UK.
Despite repeated assurance from Theresa May ruling out the prospect of a sea border, the DUP wants more certainty.
That is why the party is supporting the amendment by Tory Brexiteer Jacob Rees-Mogg to write that assurance into law.
That would effectively kill off the prospect of Northern Ireland being treated differently than the rest of the UK after leaving the EU.
But it is not clear if the government would support such a move even though it relies on the DUP to get its Brexit legislation through parliament.
The Customs Bill is due to be debated on Monday afternoon.
On Sunday, DUP MP Sir Jeffrey Donaldson said the prime minister’s Brexit negotiating plan has left her with “no room to manoeuvre”.
Mrs May – who relies on the DUP to win key votes – has urged her critics to support her plans.