The Ipsos Mori survey found support was highest among those in Northern Ireland (66 per cent) and Scotland (56 per cent), while a majority of those in England and Wales also believe the party should be able to hold another ballot (51 per cent).
In other findings from the poll, more than half think the UK will not exist in its current form in a decade.
The findings show 51 per cent across the UK would support a second referendum if the SNP wins a majority, with 40 per cent saying they would not and 8 per cent saying they do not know.
Emily Gray, managing director of Ipsos Mori Scotland, said: “The Scottish Parliament elections on May 6 look set to be a critical point in the future of the Union.
“Should the SNP win a majority of seats, as looks likely if current levels of support hold, it will be much more difficult for the UK Government to refuse a second referendum on independence.
“And these figures suggest that on balance, the UK public are on board with that course of action – more believe that the UK Government should allow a second referendum in the event of a SNP majority than say it should not.”
The survey included a representative sample of 8,558 people over 16 in the UK.
Interviews were conducted online between April 1 and 7.
Half of the UK public said they would prefer Scotland to vote against becoming an independent country if another referendum was held, while 17 per cent would prefer them to vote for this.
Results show opinion is split in Scotland – 46 per cent would prefer their country to vote against independence while 45 per cent would prefer Scots to vote for it.
Those in England and Wales are most likely to want Scotland to vote against leaving the UK – 51 per cent and 57 per cent respectively.
SNP depute leader Keith Brown said: “Each of the four nations of the UK are united in recognising the democratic wishes of the people of Scotland, but Boris Johnson and Keir Starmer continue to be at odds.”