The time has come for "megaphone diplomacy" between unionists and the Irish government over Brexit to end, the DUP's chief whip has said.
Sir Jeffrey Donaldson said Dublin has a "key role" to play in helping avoid a hard Irish border.
Prime Minister Theresa May is to set out a way forward on Monday, after MPs rejected her Brexit plan.
The backstop is an insurance policy to avoid the return of a hard border between Northern Ireland and the Republic of Ireland, if another solution cannot be found through a wider trade deal by the end of the transition period in December 2020.
If it took effect, it would mean Northern Ireland staying aligned to some EU rules, while the rest of Great Britain did not, and would place extra checks on some goods coming into NI from GB.
The DUP has dismissed the backstop, saying any proposal that leads to differences between NI from the rest of the UK poses a risk to the integrity of the union.
However, the Irish government has repeatedly said it will not support a withdrawal agreement that does not contain the backstop.
On Saturday, the Irish foreign minister Simon Coveney tweeted that his government's commitment to the withdrawal deal - and the backstop - was "absolute".
As #Brexit dominates news coverage, no surprise that some analysis today gets it wrong. I can reassure you the Irish Govt’s commitment to the entire WA is absolute - including the backstop to ensure, no matter what, an open border between Ire + NI and the #GFA are protected. pic.twitter.com/ToAt6o0B6P— Simon Coveney (@simoncoveney) January 19, 2019
Sir Jeffrey said his party's concerns about the backstop needed to be addressed, but that "unionists are ready to engage".
"The time for megaphone diplomacy on both sides is over," said the senior DUP MP.
The term megaphone diplomacy refers to the practice of one country making strong or threatening statements to make another country do what it wants.
PM in 'listening mode'
In a series of tweets, Sir Jeffrey added: "Any solution must respect the integrity of both the United Kingdom and EU but also the progress made in developing relationships on these islands.
"This is above all about future relationships and we need to show the political maturity that the challenge demands of us - all of us."
Other Stormont parties that supported the backstop, including Sinn Féin and the SDLP, have criticised Mrs May for what they described as a lack of engagement with pro-remain parties in Northern Ireland.
The prime minister will return to Parliament on Monday to make a statement on what she intends to do next.
She has said she wants to work with all parties across the Commons to reach a consensus on how to break the Brexit deadlock