The Scottish government set out plans for a radical rethink of the country’s regional authorities last month.
A new two-tiered system would see the 32 local authorities replaced with six new regional authorities.
They would have power over major issues such as economic growth and education.
Below that, between 120 and 160 community authorities would govern more local issues, such as licensing and town centre improvements.
It would be the most significant change to Scotland’s regional makeup since the current system was introduced by the UK government in 1996.
Burnham acknowledged that the Scottish government was working on local government reform, but said this had to “move faster”.
It is not clear exactly what powers might be exercised at regional level other than employment support.
Burnham appears to expect the Scottish government to devolve some of its own powers to regional authorities.
But he made clear that he will not impose a model of regional devolution like the metropolitan mayors that already operate in England.
Instead he wants to try and agree a way forward with further discussions when the prime minister sits down with all the devolved first ministers at the end of this month.