The DUP has held out the prospect of supporting a customs union as talks continue between Theresa May and Jeremy Corbyn to break the Brexit deadlock.
Sir Jeffrey Donaldson made the suggestion to BBC News NI on Wednesday evening.
It came as the Tory and Labour leaders agreed a “programme of work” to try to find a way forward to put to MPs.
Earlier, the DUP called the prime minister’s handling of the overall Brexit negotiations “lamentable”.
On Wednesday evening, Sir Jeffrey told BBC Newsline: “We would have preferred a form of Brexit that enables the UK to negotiate new trade agreements with other countries.
“That’s part of the reason for Brexit and maybe a customs union might be a temporary staging post towards that objective.
“We will wait to see what the prime minister brings before Parliament but we are very clear, we want a Brexit that delivers for all of the United Kingdom and that keeps the United Kingdom together – that is our objective.”
The DUP MP earlier told BBC Radio Ulster that regardless of what emerges in the coming days, the DUP’s stance on the union was “un-persuadable” and they remained in an “influential position” because of the government’s fragile working majority in Parliament.
MPs have been debating legislation which would require Mrs May to seek an extension to Article 50 and give the Commons the power to approve or amend whatever was agreed.
Analysis: Enda McClafferty, BBC News NI political correspondent.
Supporting one union to secure another. Might this be the new DUP tactic?
First, Nigel Dodds said he would rather remain in the EU than risk the union.
Now the party whip is saying a customs union “could be a temporary staging post” to the “preferred form of Brexit”.
On the same day, the Attorney General Geoffrey Cox suggested he, too, could live with a customs union if it helped deliver Brexit.
For the DUP, it’s about preserving the union first, delivering Brexit second.
But support for a customs union alone won’t remove the need for a backstop in the Withdrawal Deal.
Therein lies the problem; positions are shifting slowly and a softer Brexit could be inching closer.
Meanwhile, Attorney General Geoffrey Cox said it was an “article of faith” that the UK must leave the EU to honour the referendum result.
The comments come a day after Mrs May said that she will ask the EU for a further extension to Brexit.
Talks between the prime minister and Jeremy Corbyn on Wednesday were said to be “constructive”.
It is understood that each party has appointed a negotiating team, and they are meeting before a full day of discussions on Thursday.
Mr Corbyn had said he was “very happy” to meet Mrs May, and would ensure plans for a customs union and protection of workers’ rights were on the table.
The DUP has supported the government in a confidence-and-supply pact since June 2017, after a snap general election.
But it is at odds with the prime minister and her Brexit deal, because of the Irish border backstop in the withdrawal agreement.
The party opposes the plan because if it took effect, it would lead to trade differences between Northern Ireland and Great Britain, which the DUP said poses a risk to the integrity of the union.
The UK is still scheduled to leave the EU on 12 April, unless the EU agrees to another extension.
But it is likely to demand that the UK takes part in European elections, which are due to take place on 23 May.
However, Mrs May said she wanted any further extension to be “as short as possible” – before 22 May so the UK does not have to take part in the elections.
Both the UK and EU have continued preparations for a no deal, in the event that a breakthrough cannot be reached in time.