The Scottish Government should also have the "final say" on when a referendum takes place and the question asked, according to pollsters Survation.
The SNP Government recently published legislation at Holyrood to pave the way for a second referendum to be held. But control over the constitution lies with Westminster and all the contenders in the Tory leadership race have ruled out agreeing to such a vote.
Today's poll find 58% of Scots say the Scottish Government should have control over the issue, while 42% said it should be the UK Government. It was commissioned by the pro-independence Progress Scotland. Factoring in the 13% who were undecided, the polling still found half (50%) of Scots said Holyrood should be in charge, while 36% said it should be Westminster.
Former SNP Deputy leader Angus Robertson, who is managing Director of Progress Scotland said: “Clearly there is a strong majority for Scotland’s future to be decided in Scotland and not by the UK government.
"Progress Scotland is particularly interested in the changing views of people in Scotland in relation to independence.
"We are about to begin a round of focus group research to better understand what is is making the open minded and undecided change their thinking.”
The poll of 1022 Scots between June 4-7 is the first to be conducted since the Scottish Government published legislation on 29th May 2019 on holding a repeat of the 2014 which saw Scots vote against independence.
It also found 58% of Scots said the Scottish Government should have control over the timing of a second referendum, while 42% said it should be Westminster. There were 13% of don't knows, which again meant 50% wanted Holyrood to decide, while 36% said it should be Westminster.
On the the wording of the question asked in a second independence referendum, 62% said it should be the Scottish Government, while 38% said it should be Westminster. There were 14% who were undecided, meaning 54% backed Holyrood to have the final say while 33% said it should be Westminster.
Mark Diffley, independent polling advisor to Progress Scotland said: “Progress Scotland’s latest poll shows that a significant majority of Scots believe that the Scottish government should have the final say on the timing and content of a second independence referendum.
"Of those who have an opinion, nearly 6 in 10 Scots (58%) think the Scottish government, rather than the UK government, should the final say in deciding whether indyref2 should take place and on its timing, while 62% think it should decide on the question that should be asked.
“These results reflect evidence from other surveys which suggest that Scots trust the Scottish government more than the UK government and that the majority want the Scottish government to have most influence over the way Scotland is run."
The survey shows that over 80% of those who voted ‘Yes’ in 2014 think the Scottish government should have the final say on all of these issues.
But a significant minority of 2014 ‘No’ voters also back these propositions, including 28% who think it should decide on whether indyref2 should be held, 27% who think it should decide its timing and 33% who think it should have the final say on the question posed.