The organisation, which represents London's 33 councils, has switched from issuing grants to commissioning services.
At the first of a series of meetings last week to explain the new criteria, Jonathan Gore, commissioning project manager at London Councils, said the new system would lead to a "more even and fair spread of benefits".
But a senior executive at a disability organisation, who declined to reveal her name, said she was alarmed that the funding priority groups excluded disabled people.
"Groups are in danger of being shoehorned into finding services that fit the funding criteria, rather than the other way round," she said.
The four-phase commissioning process for four-year contracts opened last week.