Mr Kenny said the all-Ireland team should take on England every two years to raise money for children.
Patrick Nelson, chief executive of the Irish FA, said, however, there was no appetite in his organisation for such a team.
“That’s not going to be something that is on our radar,” said Mr Nelson. “It has not been in the past and it won’t be in the future.”
Mr Kenny was speaking in Armagh at a conference on sport boosting reconciliation in the North, and he said the matches could be played at Wembley or the GAA’s headquarters at Croke Park in Dublin and could help fund research and facilities at specialist hospitals in Belfast and Dublin.
“Wouldn’t it be a wonderful thing if we could have an all-Ireland soccer team playing England, perhaps on a biennial 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,” said Mr Kenny.
“It would be a gesture from the sports people that would have a profound impact. We could take on the might of England entirely for charity, for the children’s hospitals for the children of the island, for research and development of what can impact on their little lives.
“It is just a thought, it might be something that could become a reality.”
Mr Kenny was asked whether he would be involving British prime minister David Cameron in the proposal, but replied: “The prime minister is a very engaging person but this is for the sporting organisations.”