The monarch's 500-year-old skeleton was identified earlier this month after it was un-covered during an archaeological dig at a council car park in Leicester last year.
The remains are due to be re-interred at Leicester Cathedral next year despite campaigns to bring them to York.
Yesterday, 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 in the city.
They said in a statement: "We, the under-named, do hereby most respectfully demand 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. We believe that such an interment was the desire of King Richard in life and we have written this statement so his wishes may be fully recognised and upheld."
They added: "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 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 search for the king's remains was orchestrated by Philippa Langley, an Edinburgh-based screenwriter and secretary of the Scottish branch of the Richard III Society. Experts from Dundee University led the facial reconstruction of the monarch.
Richard grew up at Middleham Castle in the Yorkshire Dales and visited York several times during his 26-month reign.
Earlier this month, city leaders in York said they were to write to the Queen in a bid to get Richard's remains returned to his "spiritual home". A petition calling for him to be reinterred at York has been signed by more than 23,000 people.