The Labour leadership contender called on the Nationalists to work in partnership with Labour to renationalise the entire rail network including stations and ticketing.
In exchange, the Scottish Government would have a place on the board of the new National Rail along with the Welsh Assembly Government.
The proposal is a reversal of the privatisation of the railways under John Major in the 1990s which also saw the network broken up geographically and under Tony Blair’s Labour Government devolved to Scotland and Wales.
A spokesman for Burnham said: “The SNP are very good at telling people they are leftwing and strongly support the public sector and helping the poorest but when it comes to the test they are more often than not found wanting.
“Last year they could have put ScotRail back into public hands but instead took the Tory route of handing it over to a private sector company from abroad.
“What Andy Burnham is asking them now is to live up to their rhetoric and support public ownership in the best interests of people across the whole of Britain.
“That would involve supporting the renationalisation of the railway to be controlled by a new National Rail.
“The SNP would of course have to put aside their hatred of cross-Border cooperation.”
However, he added: “Of course in the discussions which would follow Andy Burnham would be very open to the idea of the Scottish Government to have a place on the board of the new National Rail.”
Burnham’s challenge comes as the Labour leadership candidates seek to answer the question on how the party can recover from the humiliation at the last election in May when they lost 40 of their 41 Scottish seats to the SNP.