Andy Burnham has again told Scotland’s First Minister that a second vote on Scottish independence is “off limits” – despite claims from John Swinney that there had been a “big shift” in stance from the Prime Minister.

The SNP leader had welcomed comments by the Labour leader when he suggested to MPs another referendum could be held if public opinion showed a demand for it.

And he wrote to Mr Burnham on Thursday, calling for talks on how an “agreed legal mechanism” for such a vote could be agreed.

That letter came after the Prime Minister indicated that Scotland should be in “exactly the same situation” as Northern Ireland – where the Good Friday Agreement sets out that a border poll can be held when the Northern Ireland Secretary determines there is “a clear consensus in the country, when public opinion has changed to the point where the call has to be considered”.

He also told the Commons he was “not aware that there is a majority of public support for another referendum” – a point he made in a letter to Mr Swinney.

On Thursday the Labour leader said he had been “simply highlighting that there is no consensus for an independence referendum in Scotland”.

And he told the Scottish First Minister: “As I made clear to you personally when we met recently in Glasgow, independence and a referendum are off limits as it would take our focus away from growing the economy and helping families with the cost of living.”

However Mr Swinney told the Press Association the Prime Minister should not be allowed to backtrack on the issue.

“I think when a Prime Minister sets out his view to the House of Commons, I think that should be taken as the most authoritative statement of a Prime Minister,” he said.

“The Prime Minister did that. He indicated just as there is a mechanism in Northern Ireland to enable the people of Northern Ireland to decide their own constitutional future, so there should be a process for Scotland doing the same.

“I welcome that, it is a big shift in Andy Burnham’s position, a big and welcome shift to recognise the people of Scotland are the ones who should determine their own constitutional future.”

The SNP leader added it was now important for the issue of support for a fresh vote on Scotland’s place in the UK to be “tested sooner rather than later”.

In his letter to Mr Burnham the First Minister stressed he would be “very pleased to engage in discussions with the Prime Minister on the details of what a mechanism would look like to enable the people of Scotland to decide their own constitutional future”.

And speaking about support for Scottish independence, he said: “We have at this very moment the largest number of independence-supporting MSPs in the Scottish Parliament, the largest number of MSPs supporting the concept of a referendum on Scottish independence.

“We also have most of the opinion polls over the last few weeks and months indicating that the preferred constitutional option of people in Scotland is independence.

“So I think we have got a number of indicators of the mood of the people of Scotland, and I think it is important that question is tested and tested sooner rather than later.”