Distant relatives of the monarch are asking judges at London's High Court to rule that Justice Secretary Chris Grayling is under a legal duty to allow a consultation "and do it fairly".
Richard's bones were discovered under a council car park in Leicester after a search instigated by Edinburgh-based screenwriter and secretary of the Richard III Society Philippa Langley. The current plan is for his remains to be reinterred at Leicester Cathedral.
The Plantagenet Alliance Ltd, formed by the distant relatives, want the remains to be buried at York Minster, claiming that was the wish "of the last medieval king of England". But their counsel Gerard Clarke told London's High Court the alliance would be satisfied with a wide-ranging public consultation exercise on where the king's final resting place should be.
Mr Clarke told Lady Justice Hallett, Mr Justice Ouseley and Mr Justice Haddon-Cave views should be obtained from the Crown, and groups including English Heritage, relevant churches, public bodies "and those who claim a family relationship with the king".