Burnham's comments, which have been welcomed by trade union Unite, have significant implications for charities currently delivering health services on behalf of government.
His views are being seen as a reversal of the recent trend towards opening up the provision of state services to not-for-profit groups.
"It's profoundly worrying in terms of the third sector's ability to compete on an equal footing," said Peter Kyle, deputy chief executive of Acevo. "You can't have equality and a level playing field when one provider is preferred. Charities have to be sure when they bid that they have a reasonable chance of winning."
Kyle, Simon Blake, chief executive of sexual health charity Brook, and Jeremy Swain, chief executive of homelessness charity Thames Reach, have signed a letter to Burnham organised by left-wing pressure group Progress.
It says: "Now is not the time to alienate important sectors that hold much goodwill and are potentially the key to the citizen-centred, high-quality health services of the future."
Unite yesterday congratulated Burnham.
"The not-for-profit sector cannot provide the stability needed for health provision due to the short-term nature of its funding, its high staff turnover and its race to the bottom on staff pay and terms and conditions," said a statement signed by Rachael Maskell, national officer for the not-for-profit sector, and Karen Reay, national officer for the health sector.
"As we have seen in the provision of services for children, charities do not share best practice as they compete with each other over contracts.
"We believe that charities do have a place in the health service, but as add-on services such as Macmillan Cancer Support's Macmillan nurses."