Warning after poll shows just 36% of Britons hope NI stays in the UK
The Ulster Unionists warned there was "no room for complacency" as an opinion poll showed only a third of people in Britain hope Northern Ireland will vote to remain in the UK.
The party said that the DUP's association with the "ultra wing of the Tory party" was not doing unionism any favours.
However, all three main unionist parties here welcomed the fact that significantly more people in the poll wanted Northern Ireland to remain in the UK than to join the Republic.
Some 36% of those questioned in the Ipsos MORI poll for King's College London hoped Northern Ireland would vote to stay in the UK in a future border poll.
A fifth said they would prefer it to leave and join the Republic.
One in three thought Brexit made Irish unity more likely over the next decade.
Roger Mortimore, professor of public opinion and political analysis at King's College, said: "When Scotland voted on whether to become independent in 2014, there was a clear majority among the public in the rest of the UK that hoped it would choose to stay. But many fewer Britons, it seems, would mind if Northern Ireland decided to leave the Union."
Almost half of Tory voters (49%) hoped Northern Ireland would vote to stay in the UK, compared to 34% of Labour supporters and 27% of Liberal Democrats.
Those who voted Remain were more likely (23%) to want Northern Ireland to leave the UK and join the Republic than those who voted Leave (15%).
Three in four adults in Great Britain have never visited Northern Ireland, and 17% have visited only once or twice. Just 3% have lived or worked here.
UUP MLA Steven Aiken said the poll showed there was "no marked call for the end of the Union".
But he added: "There is absolutely no room for complacency. The DUP's association with the ultra wing of the Tory party and the almost daily resorting to undiplomatic language by the ERG wing of their party isn't doing unionism any favours."
DUP MP Sir Jeffrey Donaldson said: "The poll shows clearly that support for Northern Ireland remaining within the Union is twice that expressed for a united Ireland. Arlene Foster has outlined the need to put forward a positive case for the Union and there is a need for unionists to ensure this case is made right across the UK.
"The issue facing people at present is not Northern Ireland voting to weaken the Union, but the insistence upon a backstop which would place an internal border inside the UK.
"There are many people from all political viewpoints who do not support such a proposal."
TUV leader Jim Allister said the constitutional future of Northern Ireland rested "exclusively with the people of Northern Ireland". He added: "Only republican dreamers think unionist EU remainers will metamorphose into republicans. I don't think so. Unionists are not republicans in waiting."
Alliance deputy leader Stephen Farry said the Good Friday Agreement made clear that Northern Ireland's future was not an issue for the electorate in Britain.
But he said the poll should "serve as a message to the political leaders of Northern Ireland that they cannot simply rely on the goodwill of others and need to ensure they do their part to make this region work".