The bones of Richard, a controversial monarch demonised by Shakespeare, were dug up last year outside a municipal building in Leicester, central England.
The University of Leicester, whose archaeologists found the site, have claimed the remains as their own and planned to rebury them at Leicester Cathedral, a decision endorsed by the government.
However, descendants of the king and other campaigners want him buried in York, the northern city which formed his power base and gave its name to Richard's family.
In a ruling on Friday, High Court Judge Charles Haddon-Cave gave the relatives permission to challenge the burial plans, saying they should have been put out to public consultation.
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"The archaeological discovery of the mortal remains of a former king of England after 500 years is without precedent," he said in a written ruling.
"In my judgement, it is plainly arguable that there was a duty at common law to consult widely as to how and where Richard III's remains should appropriately be re-interred.
Richard, vilified in Shakespeare's play as a murderous hunchback, was killed at the Battle of Bosworth in 1485 and buried without ceremony in nearby Leicester.
His death brought an end to the War of the Roses, the civil war between the families of Lancaster and York named after their respective heraldic symbols of the red and the white rose.
The judge recommended an independent advisory panel be set up to consult and decide on where the remains should be re-buried, in a bid to end the "unseemly" row.
"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.