The council's cabinet will discuss the moves on 10 January as it looks to find savings of £28m in the next financial year, including £6.95m in children's services. It anticipates making cuts totalling £65m over the next three years.
In documents to be discussed by cabinet, officers say the council can only afford to directly fund six centres.
Instead, the council hopes that private providers and local schools will take a greater share in meeting the running costs of centre services.
A council spokesman said that the anticipated model for children's centres is for six "hub" centres with the remaining centres acting as "spokes" where other organisations provide more local provision. The council also plans to open another centre to bring the total to 16, he added.
Councillor Helen Binmore, children's services cabinet member, said that local organisations already share running costs at a number of centres.
She said: "We want to avoid having to close any centres and our initial discussions with other local organisations suggests that, by working creatively together, we will be able to find significant savings from sharing running costs and by reviewing the range of services on offer at each centre."
There will also have a greater emphasis on targeting the most vulnerable families in centres rather than on offering a universal provision.
Documents presented to cabinet warn that cuts to services for vulnerable families "would almost certainly lead to a direct rise in volumes of looked-after children, raising costs elsewhere".
The cuts to children's centre funding are part of a wider restructure across the council's family support services.
Three neighbourhood teams will be created in the north, centre and south of the borough. The restructure will lead to a loss of around 50 posts. If approved by cabinet, a 90-day consultation period among staff surrounding redundancies will be launched.
Claire McCarthy, director of public affairs at 4Children, said: "Given the grant funding for Sure Start children’s centres from central government has been protected in cash terms, parents will be expecting local authorities to continue to provide these vital local services.
"Efficiency savings that have to be made from children and family services must come from back office functions or through the re-shaping of services to allow them to deliver more for less. Simply shutting centres or expecting community groups to run them without the necessary on-going resources is unsustainable and risks leading to a dramatic diminution of support."
