The poll conducted by Lord Ashcroft, the former Conservative Party chairman, found that 63% of Leave voters across Great Britain would prefer to leave the EU even if it meant Scotland, Wales or Northern Ireland leaving the Union.
Leave voters put quitting the EU above maintaining the Union regardless of their political views, with 73% of Conservative supporters saying Brexit was more important than maintaining the UK, while among Labour supporters, 50% put leaving the EU first compared with 23% who picked the Union.
Lord Ashcroft's poll also asked about the contentious issue of the Irish border, with respondents saying it was more important to leave the EU customs union than avoid a hard frontier between Northern Ireland and the Republic.
Overall, 32% agreed that it was more important to avoid creating a hard border, "even if this meant the UK not being able to do its own trade deals with non-EU countries", while 41% said the UK should leave the customs union "even if this means a hard border with customs checks".
However, there was a wide gap between Leave and Remain voters, with 55% of those who chose to stay in the EU saying a soft border was more important than leaving the customs union, while 66% of those who backed Brexit saying they would accept a hard border.
Across Great Britain, 62% of respondents said Brexit negotiations were proceeding too slowly, with 30% blaming the delays on "politicians in Britain who want a soft Brexit or to stop Brexit altogether".