Research by Britain Thinks, carried out on behalf of Oil & Gas UK (OGUK), found 23 per cent of people agreed with such a move.
However, half opposed it, while 18 per cent neither agreed nor disagreed.
Of those who agreed, 58 per cent would not maintain their position if such a shutdown meant most of the UK's oil and gas would be imported from overseas.
Elsewhere, 55 per cent agreed it would not be possible to stop using oil and gas in the UK without significantly damaging the economy.
And 56 per cent agreed that oil and gas are essential for the foreseeable future.
But the same percentage agreed oil and gas companies cannot be trusted to help get the country to net-zero emissions.
And more Scots (33 per cent) feel unfavourable towards the industry than favourable (22 per cent), with 41 per cent neutral.
Deirdre Michie, OGUK’s chief executive, said the organisation must do more to "build trust in our industry".
Scottish Greens climate spokesman Mark Ruskell said: “No one is proposing an urgent shutdown of all operations, even if that is how the Tories like to scaremonger over the issue.
"However, the fact is that overwhelming science, the UN and International Energy Agency are clear that the transition away from fossil fuels must be underway, and that expanding production is not possible if we are to have a future.”
Almost 200,000 jobs are supported by the UK offshore oil and gas industry.
However, the industry’s future is under increasing scrutiny given the impact of the climate crisis.
Britain Thinks polled 1,016 over-18s in Scotland between November 12 and 18.
Three-quarters agreed that oil and gas companies in Scotland play a vital part in the UK economy.
And 70 per cent agreed they would prefer to use UK-produced oil and gas over imports.
Respondents were asked: "Some people have called for the urgent shutdown of oil and gas production in the UK. To what extent do you agree with this view?"
Just 5 per cent strongly agreed, but 18 per cent said they would "somewhat agree". A quarter somewhat disagreed and 26 per cent strongly disagreed.
Elsewhere, 47 per cent of Scots agreed oil and gas companies in Scotland negatively impact on the UK's environmental and climate change goals, while 15 per cent disagreed.
Ms Michie said the results “contain both good news and some important lessons for our industry”.
She added: “The good news is that most people support our industry and prefer their energy to come from UK sources. There is also a strong feeling against reliance on imports.
“However, most of the UK’s existing oil and resources will be depleted by 2030 and the move to low carbon energy will take at least two decades longer.
"OGUK’s view is that the best way to minimise imports, and retain Scottish jobs and skills, is by opening new oil and gas fields.
"That would help to supply the energy we’ll need during that transition.”
Ms Michie added: "We must do more to build trust in our industry and to explain its key role in creating the UK’s low-carbon future.”