The Turnbull government has been accused of turning its back on Indigenous people after cabinet rejected the proposal for a constitutionally enshrined First Nations "Voice" based on fears it would be seen as a third chamber of Australia's Parliament.

Constitutional lawyer and Referendum Council member Megan Davis accused the government of "irresponsible" behaviour and Labor senator Patrick Dodson said the decision was disrespectful towards Indigenous people.

"There's nothing honourable about what's happened here," Senator Dodson said on Friday.

Cabinet knocked back the idea of an Indigenous "Voice to Parliament" - the central recommendation of a recent convention at Uluru and the Referendum Council's report - because it was not "desirable or capable of winning acceptance in a referendum". Mr Turnbull has now moved to develop a more modest proposal.