The survey found 64.8% of those questioned wanted the Scottish Government to produce a "Plan B" before September's referendum.
The research, for the Daily Mail newspaper, was carried out after Chancellor George Osborne last week ruled out doing a deal to allow a separate Scotland to share the pound with the rest of UK.
Labour shadow chancellor Ed Balls and Liberal Democrat Chief Secretary to the Treasury followed suit by rejecting a currency union.
In the wake of that, polling company Survation questioned 1,005 people aged 16 and over on the issue on Monday and Tuesday of this week.
A fifth of those questioned (20.1%) did not believe the Scottish Government required to draw up an alternative to its plans for a currency union, while 15.1% said they did not know.
The same poll also put the gap between support for independence and support for the Union at just nine points.
A total of 46.6% of those questioned wanted Scotland to remain in the UK, while 37.7% were backing independence, with 15.7% undecided.
Deputy First Minister Nicola Sturgeon said the poll was "exceptionally encouraging" for the Yes campaign.
She said: "It is clear that there has been a severe backlash to George Osborne's bluster and threats on the pound - with more than half of the No campaign's lead wiped out in just three weeks, and far more people more likely to vote Yes on the back of the Westminster establishment's attempted bullying rather than No."
Ms Sturgeon went on: "The pound is Scotland's currency every bit as much as the rest of the UK's, and what this poll shows is that the Scottish Government's policy for a formal sterling area between an independent Scotland and the rest of the UK is the clear preference of the people of Scotland - with 55% support when 'don't knows' are excluded.
"The Scottish Government's expert Fiscal Commission Working Group - comprised of internationally respected economists, including two Nobel Laureates - examined all of the possible currency options and concluded that a currency union is best for both Scotland and the rest of the UK.
"And a formal sterling area is more than twice as popular as the next option in terms of what people in Scotland think will happen after independence.
"What this poll illustrates above all is that the people of Scotland are calling the bluff of George Osborne, Ed Balls and Danny Alexander - whose actions last week have given a big boost to the Yes campaign."
Blair Jenkins, chief executive of the pro-independence Yes Scotland campaign, said the research "continues the trend we are finding of an ever-narrowing gap in voting intentions - now down to just single figures, and needing a swing of less than five points to put Yes ahead".
He added: "The fact that the Survation poll was carried out in the wake of Chancellor George Osborne's visit to Edinburgh, with his threatening bluster on the pound, merely mirrors our findings of recent days that many more voters are coming to us to declare their support since last week.
"All the evidence we have is that the warnings and threats delivered by Mr Osborne and his Westminster allies in Labour and the Lib Dems have backfired badly on the No campaign."
A separate poll which was also published today put the Yes vote at 29%, with 42% planning on voting No, and 29% undecided.
The TNS poll was carried out in the first week of February, before the Tories, Labour and Liberal Democrats all ruled out a currency union.
TNS head in Scotland Tom Costley said: "The debate on independence has stepped up a gear in the last few days.
"Opinion in Scotland showed little change over the last few months, with a relatively high level of don't knows, and our polls have consistently shown that people are looking for more information on which to make their decision.
"It may be that the recent activity, especially the firm statement by the anti-independence parties on the future of the pound, will stimulate more debate, bring out more information and help undecided voters to make up their minds."