Cabinet knocked back the central recommendation from the non-partisan Referendum Council and Prime Minister Malcolm Turnbull is moving to shore up support for a more modest proposal, The Courier-Mail reported on Thursday.

A spokesman for the government told Fairfax Media it "continues to carefully consider the Referendum Council's call to amend the constitution to provide for a national Indigenous representative assembly to constitute a 'Voice to the Parliament'".

Opposition Leader Bill Shorten and Senator Patrick Dodson said "this looks to be a sad day for First Nations Australians and all of us who believe First Nations people deserve meaningful recognition in our nation's constitution".

"All of this indicates that Malcolm Turnbull is seeking to impose his own view of what recognition of First Nations people should be and does not seem willing to listen to First Australians about what recognition and reconciliation means to them," the pair said in a statement.

They warned that "without bipartisanship, the path from here is far from clear".

Fairfax Media has confirmed Indigenous Affairs Minister Nigel Scullion and Attorney-General George Brandis have co-sponsored a submission to cabinet on the referendum.

Asked if the Voice proposal had been rejected, a spokesman said Senator Scullion would not comment on cabinet deliberations.

While Labor has backed the constitutionally-enshrined Voice, as well as a Makarrata truth and reconciliation commission, Mr Turnbull has publicly expressed caution about overly ambitious referendum proposals.

Earlier this week, the Law Council of Australia threw its "full and unqualified support" behind the Referendum Council's recommendations.

"The Law Council considers the constitutional reforms recommended by the Referendum Council to be a necessary and important step towards Aboriginal and Torres Strait Islander peoples' self-determination," president Fiona McLeod SC said on Tuesday.