The games could be played at Wembley in London or the Gaelic Athletic Association's headquarters of Croke Park in Dublin and could help fund research and facilities at specialist hospitals in Belfast and Dublin.
Mr Kenny said the idea could be attractive to everyone.
"It would be a gesture from the sports people that would have a profound impact," he added.
The Queen visited Croke Park in 2011 as part of a trip aimed at helping to reconcile Ireland and Britain.
The 82,000-seat stadium in the north of the Republic's capital hosts the final stages of the GAA's football competitions between different counties and is a key part of Irish national identity.
Mr Kenny was in Armagh City at a conference on sport boosting reconciliation in Northern Ireland.
He said the Irish Government was building a hospital for all the children of the island of Ireland.
"We could take on the might of England entirely for charity, for the children's hospitals (of Belfast and Dublin), for the children of the island, for research and development of what can impact on their little lives," he added.
"It is just a thought, it might be something that could become a reality."
He was asked whether he would be involving Prime Minister David Cameron.
"The Prime Minister is a very engaging person but this is for the sporting organisations.
"The sport is about the children of the island. Wouldn't it be a wonderful thing if we could have an all-island soccer team playing England, perhaps on a bi-annual basis, playing in Ireland and England, the purpose of which would be the development of research and the development of both children's hospitals on the island of Ireland."
He said it could be a unique integration of sport and services for the children of Ireland.