The intervention from nine Welsh MPs comes after Dr Jeffrey John (above), Dean of St Alban's Cathedral, accused the Church in Wales of homophobia after he was not selected to become Bishop of Llandaff at a meeting of existing bishops, which aimed to pick a new one, last week.
Dr Jeffrey John has been in a civil partnership since 2006 but said at the time he was celibate with his partner - an arrangement which does not break Church in Wales teaching. Gay marriage is against its law.
The five bishops were required to select a nominee with a two-thirds majority but could not. Given that the selection process for bishops is strictly confidential, it is not known for sure how much support Dr Jeffrey carried.
In a letter, he claimed: "The only arguments adduced against my appointment - in particular by two of the bishops - were directly related to my homosexuality and/or civil partnership - namely that my appointment would bring unwelcome and unsettling publicity to the diocese.
"I put it to you that this is not a moral or legal basis on which to exclude me."
Now Welsh MPs spearheaded by Bridgend's Madeleine Moon (below) have called for the Church in Wales to reconsider if it did refuse to select Dr Jeffrey on the grounds of sexuality.
Their open letter said: "It is with some reluctance and regret that we find ourselves needing to place our concerns relating to recent events within the Church in Wales on the public record.
"We feel that the present process has been flawed and has led to considerable disharmony, anger and confusion. We respectfully recommend that there is a pause in the appointment process to allow emotions to cool and sound council to be heard.
"It would then appear appropriate that a new election is called, open to past and new candidates to apply and an open and transparent decision be made.
"There is now a need for healing and rebuilding of trust and confidence in the legality and moral leadership of the appointment of a much needed new Bishop of Llandaff and we hope this will soon be possible."
Dr Barry Morgan retired as Bishop of Llandaff and Archbishop of Wales earlier this year
A Church in Wales spokesman said: "We understand the disappointment felt by all the candidates considered by the Electoral College who did not secure enough support to be elected as Bishop of Llandaff. However, we are satisfied that the Electoral College process was carried out properly and fairly.
"The meeting was confidential and we will not comment on speculation about the nomination and discussion of candidates. However, we strongly deny allegations of homophobia in the process. Neither homosexuality nor participation in a civil partnership are a bar to any candidate being either nominated or elected as a Bishop in the Church in Wales. Moreover, this was made clear to members of the Electoral College by its President, the Bishop of Swansea and Brecon.
"The Bishops have stressed during the whole process that whoever becomes Bishop of Llandaff, whatever their circumstances, will receive their full support.
"The Bishops are now acting carefully in full accordance with the Constitution. Unlike the Electoral College process, there is no fixed timetable for an appointment process, however, the Bishops would wish to announce any appointment made as soon as all necessary formalities are finalised. The appointment process is underway and we see no reason to halt it."
