Many of his relatives want the body buried in York, claiming it would have been the King's wish.
A High Court judge in London has given permission to the Plantagenet Alliance to bring judicial review proceedings against the Justice Secretary and the University of Leicester.
Richard was killed at the battle of Bosworth in 1485 --ending the Wars of the Roses and the Plantagenet dynasty - and was hurriedly buried in the church of the Greyfriars in Leicester, which was subsequently lost during redevelopment.
His remains were discovered buried under a municipal car park owned by Leicester City Council after a search instigated by Edinburgh-based screenwriter and secretary of the Richard III Society Philippa Langley.
Mr Justice Haddon-Cave, in granting permission for a hearing later in the year, said the discovery was "without precedent".
He said: "I would, however, urge the parties to avoid embarking on the (legal) Wars of the Roses Part 2.
"In my view, it would be unseemly, undignified and unedifying to have a legal tussle over these royal remains.
"This would not be appropriate, or in the country's interests."
Mr Justice Haddon-Cave, sitting in London, said the discovery was the result of "inspired, determined and meticulous work" by Richard III Society members, the University of Leicester Archaeological Services, and Leicester City Council.
Archaeologists from the University of Leicester were granted permission to excavate the car park and to decide where the bones should be reburied by the Ministry of Justice. Under the terms of a licence to dig up the remains, the university has decided that Richard should be re-interred at Leicester Cathedral.
But opponents argue he would have wanted to be put to rest at York Minster.
Mr Justice Haddon-Cave said: "I would strongly recommend that parties immediately consider referring the fundamental question - as to where and how Richard III is reburied - to an independent advisory panel made up of suitable experts and Privy Councillors."