A poll by Survation, carried out for the pro-UK campaign group Scotland in Union, found 58 per cent believe the UK should “retain its independent nuclear deterrent”.
Just 20 per cent said it should not retain the deterrent, while 23 per cent didn’t know.
Following any vote for independence, they would seek to remove them from the Faslane naval base on the Clyde.
The poll of 1,050 adults in Scotland, carried out between April 29 and May 3, also found around four-fifths of people in Scotland believe membership of the G7, Nato and the UN Security Council is important.
The Greens oppose an independent Scotland joining Nato.
Pamela Nash, chief executive of Scotland in Union, said: “As proud Scots, maintaining our position and influence in the world is key.
“And it’s clear from this opinion poll there is little appetite to diminish our global standing.
“SNP politicians are frequently trying to put themselves on the global stage, yet they remain driven by an insular ideology that would put an extra border between us and the world.
“They are ready to walk away from the power for good and influence that we have at the G7 and the UN, and to diminish our standing in Nato, which has been key to our peace and safety for decades.
“Scottish people want to continue to participate fully in the world, to make the most effective contributions to global institutions, and to retain the benefits of our membership.”
An SNP spokesman said: "An independent Scotland’s security and safety is best guaranteed as a non-nuclear member of Nato, just like Denmark and Norway and almost every other member."
Scottish Greens MSP Ross Greer said: “Nuclear weapons cannot discriminate between military and civilian targets. They are world-ending weapons of mass slaughter. No-one wins a nuclear war.
"Beyond that moral case though is the financial outrage of the Westminster Government spending hundreds of billions of pounds on a new nuclear arsenal whilst refusing to help families struggling through this cost-of-living crisis.”