Mr Milburn said the current system was "confused and complex".
The review, for consultants PwC, called for a gradual evolution of the structures, saying those who worked in the health service supported reform.
But the Department of Health said its plan for the future would be "delivered within the NHS's existing structures".
The PwC report was critical of the changes introduced in 2012 by then Health Secretary Andrew Lansley, which have resulted in the creation of "myriad" national organisations, including:
The consultancy said it meant hospitals and other services faced the "daunting challenge of managing competing requirements".
The review said many of the functions of these competing organisations could be gradually merged into one and in time regional bodies be created to take charge of budgets for the NHS and council-run care services.
It said this would aid integration between the two systems - seen as key to shifting care out of hospitals and coping with the ageing population.
The report acknowledged these changes would take up to a decade - but said the polling it had carried out with NHS staff suggested there was widespread support for reform.
Mr Milburn said: "Despite the best efforts of its leaders to make it work, the current national architecture is confused and complex.
"The artificial divide between health and social care makes as little sense as the division of labour between a myriad of national bodies.
"Organisational change is always a risk but without it, the move towards integrated systems will be undermined."
But the Department of Health said there was no need for more reform as there was "a plan for the future... that is being delivered within the NHS's existing structures".
It said this would be funded by the extra money being invested in the NHS during the Parliament.
The way the NHS in England is organised is hindering its ability to meet its challenges, a review led by former Health Secretary Alan Milburn says.