The Bishop of St Davids in the Anglican Church in Wales is on sick leave amid pressure on her to resign.
Dr Joanna Penberthy is at the centre of a political storm over her "never trust a Tory" tweet in March, for which she apologised and closed down her Twitter account.
"I, of course, trust and have trusted many Conservatives and know there are many honourable people in that party," she said, adding that she was sorry her tweets "may have caused upset and offence".
The US-based website, Anglican Ink, reports that the Archdeacons of her diocese have written to clergy with the news that Dr Penberthy, 61, is now off work on the advice of her doctor.
In the letter dated June 19, Archdeacons Paul Mackness, Dorrien Davies, Eileen Davies and Mones Farah told St Davids clergy: "We are writing to you today to let you know that Bishop Joanna is unwell and, on the advice of her doctor, will be away from work for the next month.
"You may be aware that correspondence between the Secretary of State for Wales, the Archbishop of Canterbury and the Bishop of Bangor was published yesterday.
"There is also an updated news article on the Church in Wales website. Please be assured that the Bench of Bishops, supported by national and diocesan officers are engaging with both Welsh and UK Governments to ensure that Bishop Joanna's apology is heard and that steps are taken to restore damaged relationships."
The Bishop of Bangor, Andy John, is the acting head of the Church in Wales after the retirement of its Archbishop, John Davies, in May.
In a statement released on June 18, Bishop John said: "Clergy have a role to play in holding politicians and others to account in their public office. In order to do this effectively, clergy need to be respectful and robust in both their delivery and discourse. The Bishop of St Davids has apologised for the hurt she has caused as a result of her posts on Twitter and the Church in Wales has welcomed her statement.
"I add my apology on behalf of the Church in Wales; I regret the intemperate language that she used and the hurt it caused. I have spoken with the bishop and believe her to be sincere in her apology and that she wishes to improve the way in which her ministry operates in the public domain.
"It is clear to me that bishops cannot make distinctions between personal and public statements when both are publicly available. It is also clear that bishops require advice and support which enables us to be effective in our ministry and it is my intention to secure this in consultation with my colleagues."
The Church in Wales is a separate Anglican Province from the Church of England. But according to The Daily Telegraph ,the Conservative Welsh Secretary, Simon Hart MP, wrote on June 8 to the Archbishop of Canterbury, Justin Welby, to complain about Dr Penberthy's "politically motivated messaging".
Archbishop Welby has reportedly written back to Mr Hart with an apology.
"I am deeply embarrassed by the use of such language by a church leader. It is absolutely unacceptable, and I am truly sorry because I am very well aware of what it is like to be trolled in this way," he said, according to The Telegraph.
Dr Penberthy was the first female bishop to be appointed by the Church in Wales. Her tweet in March said "Never, never, never trust a Tory," and was posted in response to another Twitter user who claimed that the Conservatives plan to abolish the Welsh Assembly, the Senedd.
The Telegraph quoted an unnamed Tory MP who branded her "disgracefully divisive" and "not fit to be a bishop".