The topic was raised in Westminster in response to SNP MP Chris Law.
Burnham 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.”
The prime minister added: “I would say exactly the same situation in Scotland.
“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 said despite the comments there had been no change in the prime minister’s position.
Burnham’s spokesman said: “The PM has 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.”
