John Swinney is seeking discussions with Andy Burnham over how a legal framework for a second independence referendum could be established.
The SNP leader has written to the Prime Minister after comments in the Commons indicated such a vote could take place if public demand is demonstrated.
Made during Prime Minister’s Questions, the remarks suggest a shift from the Labour leader, who previously informed Mr Swinney that a fresh vote was “off the table”.
In light of Wednesday’s statement, Mr Swinney stated it is "important" that he and Mr Burnham explore "how we turn that principle into an agreed legal mechanism, as exists in Northern Ireland".
He argued that previous UK leaders’ refusal to permit another ballot was "clearly unsustainable", describing Mr Burnham’s comments as a "long-overdue acceptance of the reality that the people of Scotland will decide their own future".
The issue came up during Prime Minister’s Questions, during which Mr Burnham compared the situation in Scotland to the conditions for a border poll in Northern Ireland.
In response to a Scottish National Party MP, Chris Law, he said the Good Friday Agreement made clear that a border poll would be “considered by the Northern Ireland Secretary when there is a clear consensus in the country, when public opinion has changed to the point where the call has to be considered”.
“I would say exactly the same situation in Scotland,” Mr Burnham added.
“I am not aware that there is a majority of public support for another referendum and until that changes there won’t be one.”
Downing Street insisted there had been no change in the Prime Minister’s position.
Mr Burnham’s press secretary said: “The PM’s been repeatedly clear on this. In his call with the Scottish First Minister in his first week, the PM stated upfront that another referendum on independence was off limits and as our election manifesto makes clear, Labour does not support independence or another referendum.”
Mr Swinney however said he was “encouraged” the Labour leader had “reconsidered” his stance.
He also told Mr Burnham: “In May, the SNP won a historic and unprecedented fifth term in office, and the people of Scotland elected more pro-independence MSPs than at any point in the Parliament’s history.”
Adding there “must be a discussion about where power lies across these islands”, Mr Swinney said: “By concluding a clear, understandable mechanism, both sides of the constitutional debate can move forward constructively in the full knowledge of an agreed process by which the Scottish people can exercise their democratic right at a time of their choosing.”
In August, the Scottish Government published the draft Independence Referendum Bill to hold a second referendum, which defines the franchise for a vote, and sets out the question the referendum would ask: “Should Scotland be an independent country? Yes or No.”
The government said the Bill will be introduced at Holyrood once the necessary powers are secured.
In the days after the SNP Government was re-elected in May, a motion in the Scottish Parliament to call on Westminster “to make a Section 30 order under the Scotland Act 1998 to devolve the powers to the Scottish Parliament to hold a referendum on Scottish independence” passed by 72 votes to 55.
Meanwhile, elections expert Professor Sir John Curtice said the Prime Minister’s comment “does read like something of a misstatement”.
Sir John said opinion polls during this year’s Holyrood election and afterwards “suggest there is at least a small majority in favour of independence”.
But speaking on BBC Radio Scotland’s Breakfast programme on Thursday, he said: “Both sides in the argument can point to evidence that appears to substantiate their case.
“Those on the nationalist side of the argument can say ‘look, there is a majority’ although it is very small, it is about 52% to 48%.
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