The monarch’s bones were discovered underneath a car park in Leicester last year, following an archaeological dig.
More than 15,000 people have now signed an online petition supporting York’s claim to have him reburied there.
Charles Brunner, a descendent of Anne of York, Richard’s sister, said: “We repatriate those who give their lives in battle, so why is this different?”
Mr Brunner, who is also related to Richard’s brother Edward IV, added: “We don’t leave them a few miles from where they fell: they are brought home. Richard III deserves this, as does anyone. It is a matter of justice.”
Richard, the last monarch of the House of York, grew up at Middleham Castle in the Yorkshire Dales and visited York several times during his short reign.
Stephen Nicolay, the 16th great-grandson of Richard Plantagenet, 3rd Duke of York (father of Richard III) said: “York and the county of Yorkshire was, and remains, the physical and spiritual home of King Richard III.
“The burial of his exhumed remains should therefore be, without question, at York Minster which was, in life, his own wish.”
The campaign comes despite the fact York Minster has rejected calls for the remains to be buried there
James Alexander, leader of the City of York Council, said support for the king “is as deeply and widely felt now as in his lifetime”:
“A number of his descendants have appealed to us to uphold their ancestor’s known living wishes to be buried in York where the king was establishing a chantry of 100 priests,” he said.
“By the time of his death in 1485, six altars were already in place in the Minster where prayers for his and his family’s souls were said.”
Mark Ormrod, Professor of Medieval History at University of York supported the relatives’ claim.
“In Richard’s own time, royal remains were often exhumed and moved significant distances for more dignified reburial,” he said
He added Richard’s own preferences and good historical precedent “should see England’s last Yorkist king” re-interred in York Minster.