The bones of the monarch, who was killed in the Battle of Bosworth in 1485, were identified this month after being discovered underneath a council car park in Leicester.
But the find, one of the most sensational in archaeological history, has since provoked a dispute over where the remains, which include the king's crooked spine, should be interred, The Telegraph reported.
After Westminster Abbey and York Minster both made claims, it was finally decided that he should be buried at Leicester Cathedral in what some said was a "finders keepers" agreement.
But now the debate has been re-opened by his descendants, who wrote an open letter demanding that his remains be buried in York in North Yorkshire, England.
Nine of Richard III's descendants said they believed the king, the last monarch of the House of York, would have wanted to be buried there.
"We, the under-named, do hereby most respectfully demand that the remains of King Richard III, the last Plantagenet King of England and our mutual ancestor, be returned to the city of York for formal, ceremonial reburial," the descendants said in a statement.
"We believe that such an interment was the desire of King Richard in life and we have written this statement so that his wishes may be fully recognised and upheld," the statement said.
King Richard III was the last King of the House of York and the last of the Plantagenet dynasty which had ruled England since the succession of King Henry II in 1154.
"We, the under-named blood descendants, unreservedly believe that King Richard is deserving of great recognition and respect and hereby agree to dutifully uphold his memory. With due humility and affection, we are and will remain His Majesty's representatives and voice," the descendants said.
Earlier this month, city leaders in York said they were to write to the Queen and the Ministry of Justice to try to get the remains returned to his "spiritual home". Richard grew up at Middleham Castle in the Yorkshire Dales. He also funded part of York's gated walls.
A petition calling for the remains to be reinterred at York has been signed by more than 23,000 people.
But the Ministry of Justice said it was the University of Leicester's decision to make because it had been given consent to exhume the body.
London: The descendants of King Richard III have called for the monarch's recently discovered remains to be buried in York, as the row over their burial is threatening to develop into a drama of Shakespearean proportions.
First Published: Monday, February 25, 2013, 16:15