Milburn, a former health secretary and the man thought to be preparing Labour's next election manifesto, said the voluntary sector should become as integral in carrying out public services as the public and private sectors.
He lauded the existing role charities play in carrying out public services in areas such as learning disability, mental health and regeneration and promised "unrivalled new opportunities for the voluntary sector to expand still further".
But his vision of a greater role for charities was met by a sceptical response from some parts of the sector, with the NCVO describing the idea of wholesale transfer of services to voluntary organisations as "dangerous".
Milburn told a conference organised by chief executives' body Acevo that where a public service is to be contracted out, the voluntary sector should have an automatic right to bid along with public and private-sector organisations.
He also said that every government department should sign a concordat with the voluntary sector setting out where charities can best carry out public services.
He hinted that Acevo's proposal for a Voluntary Finance Initiative - long-term contracts for service delivery charities - could become government policy.
But he warned that charities would have to reform. "The sector cannot have its cake and eat it," he said. "When it becomes part of mainstream service provision, accountability must follow. The voluntary sector will need to get its own house in order in terms of how it is organised and governed. In short, it needs to become more professional and less amateur."
Campbell Robb, director of public policy at the NCVO, said charities should think carefully before committing their "specialist" skills to public service delivery. "It is vital that voluntary organisations only enter in partnership with government when there is clear evidence that it will directly contribute to meeting their users' needs," he said. "Talk of new initiatives that will enable the wholesale handover of large sections of services to the voluntary sector is dangerous."
But Acevo denied that Labour's plans would see vast swathes of public services foisted onto an ill-prepared charity sector. "Neither Acevo nor Alan Milburn advocates a wholesale transfer of public services to the third sector," said head of policy Nick Aldridge. "We are calling for a more mature partnership between the state and third sector so that, where there are shared goals, we can improve results for service users."
Mencap chief executive Jo Williams said she welcomed the opportunity for an expanded role for the sector, but wanted an end to charity subsidy of public services. Michael Lake, director general of Help the Aged, said the nature of charities could change.
"They will become more like businesses, and the nature of organisations will become more commercialised rather than the heart of the charity being the desire to right some wrong," he predicted.