Of those surveyed, 37% said a poll should be held within five years, while 29% believed it should be held but not in the next five years, while 32% said a border poll should never be held.
But in the Republic, attitudes are different, with 44% saying a poll should be held, but not in the next five years.
Another 37% said it should be held in that timeframe, with just 12% saying it should never take place.
Even if a poll were to take place in the next five years, it would be likely to fail in Northern Ireland, with the same poll recently showing voters would choose to remain in the UK if the question was asked in the present day.
Of those surveyed, 49% said they would vote to stay in the UK while 43% would support a united Ireland. The remainder were undecided.
The Lucid Talk poll in Northern Ireland had a sample size of 2,845 and a 2.5% margin of error, was conducted for BBC NI’s Spotlight programme over April 5-7.
The poll in the Republic was conducted by Lucid Talk/Ireland Thinks between April 6 and 9. The sample size was 1,008 with a 2.5% margin of error.
Further results were released on Thursday night’s edition of The View on BBC Northern Ireland
It found that Brexit largely hasn’t changed attitudes to the constitutional question among both unionist and nationalist communities.
Asked if the UK’s exit from the EU had changed their mind on the subject, 45% of those who support remaining in the UK said it hadn’t.
Of those who support a united Ireland, 32% said the Brexit referendum hadn’t changed their minds.
However, people who had supported remaining in the UK were far more likely to have changed their mind as a result of Brexit.
Some 19% who had supported the Union said they would now support Northern Ireland joining a united Ireland.
That compares to just 1% of those who previously supported a united Ireland who said they now favour remaining in the UK. The remainder were undecided.
There were some stark differences of opinion between voters in each jurisdiction over what changes might be necessary to accommodate the whole island in a united Ireland.
Asked if Ireland should change its tricolour flag to accommodate unionist sentiments, 46% of Northern Irish voters agreed while 63% of voters in the Republic disagreed.
In Northern Ireland, 29% disagreed with the proposition, and 18% were neutral. In the Republic, just 21% agreed, while 14% were neutral.
Similarly, a majority of those in Northern Ireland, 57%, said Ireland should change its national anthem, Amhran na bhFiann/A Soldiers Song.
In the south 46% disagreed, although 38% agreed.
But there was some common ground, with both jurisdictions agreeing that Ireland should move to a British-style NHS system, with no charges to visit A&E, the GP or to receive prescription medicine.
In the North 82% agreed, while 70% of respondents in the Republic agreed.
People in both jurisdictions also agreed that a Scottish vote for independence in the next five years would make a united Ireland more likely.
In Northern Ireland 60% said it would be more likely, 2% said less likely and 35% said it would make no difference either way.
In the Republic, 64% said it would be more likely, 3% said less likely, and 27% said it would make no difference either way.
The survey was carried out over the Easter period when clashes were taking place in a number of areas across Northern Ireland, sparked by loyalist anger over the Northern Ireland trade protocol and the PSNI’s handling of a republican funeral during the pandemic.
The BBC commissioned the polls as part of a special programme reflecting on the centenary of Northern Ireland’s foundation.