Boris Johnson has confirmed his rejection of a Northern Ireland-only backstop as a solution to the Brexit deadlock, the DUP has said.
Deputy leader Nigel Dodds said the party had a "very good meeting" with the prime minister in Downing Street.
The DUP had insisted the government would not back such a proposal as it would not have "unionist consent".
In a statement, it said a deal without cross-community support is "doomed to failure".
In the statement, DUP leader Arlene Foster said the prime minister had "confirmed his rejection of the Northern Ireland-only backstop and his commitment to securing a deal which works for the entire United Kingdom as well as our neighbours in the Republic of Ireland".
A Number 10 spokesman said the 50-minute meeting was "positive".
The meeting with Mr Johnson came as pressure grows on Downing Street to reach a deal with the EU ahead of a key summit in October.
The prime minister has insisted he will not seek an extension to the Brexit deadline if there is no agreement with the EU.
The DUP has said it wants a "sensible deal", but will not support any arrangement that could see Northern Ireland treated differently than the rest of the UK.
In Dublin on Monday, Mr Johnson said he had an "abundance of proposals" to replace the Irish border backstop, the insurance policy to maintain an open border between Northern Ireland and the Republic of Ireland after Brexit, until a wider solution is found.
The idea of a NI-only backstop was first suggested early in the Brexit negotiations.
Former PM Theresa May rejected it in 2018 because she relied on the votes of the 10 DUP MPs in Parliament.
The Irish government has said it is willing to look at a "Northern-Ireland specific solution".
On Tuesday, Ireland's EU Commissioner Phil Hogan told RTÉ that there is "movement" happening on both sides of the Brexit negotiations.
What was the NI-only backstop?
Proposed by the EU, it would have kept Northern Ireland in the EU's customs union, which would have meant applying EU tariffs for goods entering NI from outside the EU customs union - including Great Britain.
It would also have applied EU single market rules for goods and animal products in Northern Ireland.
This would have got rid of the need for checks at the Irish border, but would have led to checks on goods passing between Northern Ireland and Great Britain.
It was criticised by the DUP and other unionist parties for "creating a border in the Irish Sea" and posing a risk to the union.
In February 2018, Theresa May said no prime minister could back such a proposal.
How does it differ from the backstop agreed by the UK and EU?
The main difference between the original backstop and the agreed solution is in relation to customs.
In the backstop agreed by the UK and EU in November 2018, the whole of the UK, would be included in a "temporary customs territory" with the EU.
Some pro-Brexit MPs claimed the backstop would be used to permanently trap the UK in the EU customs union, preventing the country from striking its own trade deals.
It would also see Northern Ireland staying aligned to some rules of the EU single market.
These arrangements would apply unless and until both the EU and UK agreed they were no longer necessary.
What has the DUP said?
Sir Jeffrey Donaldson told the BBC the proposal would "run contrary to the principles at the heart of the Good Friday Agreement that say there should be no change to the relationship between Northern Ireland and Great Britain unless there is consent for it".
"There would not be unionist consent for such an arrangement."
On Monday night, DUP deputy leader Nigel Dodds said the party was prepared to look at "arrangements" to maintain all-island agri-food regulations - but only under a previously agreed role for the devolved assembly at Stormont.
What do other parties think?
Non-unionist political parties in Northern Ireland support the backstop and would be happy if the UK and EU agreed to an NI-only backstop.
The Ulster Unionist Party also opposes a backstop in any form, and claims unionists in Northern Ireland are being "hung out to dry".
However, business and farming groups in Northern Ireland have urged support for a backstop, saying it is an acceptable way to protect cross-border trade.