Dr Morgan, who's been the Archbishop for 12 years is concerned increasing demands put the current model at breaking point and wants the Church to consider alternatives.
His main issue is that the Archbishop is elected from the diocesan bishops, with the elected person taking on both jobs. They lead the diocese as well as the wider Church.
Several efforts at reforming the office have been made over the years but none have yet been accepted. A recent independent review of the church, however, recommended creating a permanent see for the Archbishop and that recommendation has brought the issue back onto the agenda.
Speaking in his Presidential address to members of the Governing Body of the Church in Wales, Dr Morgan said, "In 1920, with only four dioceses and a more leisurely pace of life, that was fine, but four of my eleven predecessors have said that this model is at breaking point or have found the post very demanding for different reasons. I add my voice to that chorus.
"Now when five of the twelve archbishops of Wales have said to the church that this model is hard to sustain, the church needs to take that seriously... I realise the argument about giving every diocese its moment of glory, if that's what it is by having the archbishop as bishop, but I ask you to consider the cost to the holder of the office as some of my predecessors have highlighted."
He's calling for the role of Archbishop to be based in Cardiff, with the diocesan job being opened up to a new candidate.
Urging members to consider the proposal, Dr Morgan stressed its importance. He said: "I have no axe to grind because none of this will affect me but I do have a concern for the workload of whoever succeeds me."