Labour’s Stephen Kinnock says he and other MPs are involved in discussions with the DUP, to try and win their support ahead of key Brexit votes in Parliament on Monday.
MPs will vote for a second time on a series of options, amid deadlock over the government’s withdrawal deal.
Mr Kinnock is backing Common Market 2.0 – a proposal that would keep the UK closely tied to the EU.
It is not clear if the DUP will vote for any of the tabled options.
A party source said it would wait and see which options were selected by Speaker John Bercow.
They added that the party would vote in a way that stuck to its principles regarding the union, and would ensure the 2016 referendum result was respected.
Common Market 2.0, also known as Norway Plus, would propose membership of the European Free Trade Association (EFTA) and a “comprehensive customs arrangement”.
Its proponents say this would remove the need for the Irish border backstop to ever take effect.
Mr Kinnock told BBC News NI that MPs supporting Common Market 2.0 were involved in “intense discussions” with the DUP, and that he hoped they would support it in the Commons.
“They understand that frictionless trade is what deals with the backstop and means it would never have to be used and that’s why they abstained from Common Market 2.0 on Wednesday,” he said.
“I think that was a very interesting signal that the DUP can see that Common Market 2.0 is a way of leaving the European Union, but doing so in a way that means the backstop never needs to be activated.”
The SNP has said it will back the proposal, which has been tabled by Conservative MP, Nick Boles.
Monday’s votes at Westminster are non-binding, but the intention is to see what outcome, if any, commands a majority.
On Sunday evening, the DUP’s Sammy Wilson said that his party would not vote for Theresa May’s Brexit deal even if she presented it to the House of Commons “a thousand times”.
Speaking to BBC News NI, he said the party’s position was fixed.
On Friday, the DUP’s deputy leader Nigel Dodds said he would prefer to “remain in the EU rather than risk Northern Ireland’s position” in the union.
The party has stressed that the Irish border backstop, contained in the government’s deal, breaches the party’s one red line when it comes to Northern Ireland being treated differently from Great Britain.
It has argued that any differences could pose a risk to the future of the union of the UK.
The UK is now scheduled to leave the EU on 12 April, after MPs again rejected the withdrawal agreement negotiated by Mrs May.
‘DUP on wrong side of argument’
Away from Westminster, Sinn Féin leaders, Mary Lou McDonald and Michelle O’Neill, are in Brussels to meet the EU’s chief Brexit negotiator Michel Barnier.
Speaking to the BBC’s Good Morning Ulster programme, Sinn Féin’s Michelle O’Neill said the DUP “continue to be on the wrong side of the argument” in terms of Brexit.
Ahead of their meeting with Michel Barnier and Guy Verhofstadt, Ms O’Neill said, like previous engagements, they would be focused on “the need to protect the primacy of the Good Friday Agreement”.
“Member states have promised to avoid a hard border even in the context of a no-deal,” she said.
“They have a commitment to withhold the integrity of the customs union and the single market if there is a crash-out Brexit so we want to explore that with Michel Barnier – how they’re going to protect the island of Ireland, the interests of the people of Ireland and to work to ensure there’s no return of a hard border in Ireland.”
Monday 1 April: MPs hold another set of votes on Brexit options to see if they can agree on a way forward
Wednesday 3 April: Potentially another round of so-called indicative votes
Wednesday 10 April: Emergency summit of EU leaders to consider any UK request for further extension
Friday 12 April: Brexit day if UK does not seek/EU does not grant further delay
23-26 May: European Parliamentary elections