The cuts in Wales are part of the corporation's wider £500m savings plan
The first minister has warned of an "irreversible deterioration" in the way the BBC covers Wales, if the corporation proceeds with cuts as planned.
The warning comes in a letter to BBC director-general Matt Brittin, which has been seen by BBC Wales.
The letter also claims that the changes could lead to the BBC being in breach of its public service obligations under its Royal Charter.
In reply, Brittin told First Minister Rhun ap Iorwerth that the BBC takes its obligations to Wales "extremely seriously".
In September BBC Wales announced 58 job cuts as well as a 20-hour annual cut in TV programming as part of UK-wide plans to save £500m.
The letter is signed jointly by ap Iorwerth and the Welsh government's Culture Minister Heledd Fychan.
It has also been sent to the UK government's Culture Secretary Lisa Nandy, as well as Rhuanedd Richards - who was appointed Director, BBC Nations on Tuesday.
Ap Iorwerth and Fychan wrote: "It is important that we convey our concerns that if the proposed changes are implemented, they will lead to an irreversible deterioration in the way in which the BBC serves the people of Wales.
"These cuts will have major implications for the delivery of the cultural and economic objectives for which the Welsh government is responsible, especially given our desire as a government to work towards strengthening public service media platforms in Wales."
They add that they understand that the media landscape is evolving and that audience consumption is changing and "recognise" the need for "fundamental reform" of the way the BBC is funded.
But they warned: "Decisions about which services to fund are for you to make as an independent broadcaster, however, we are concerned that these changes are so significant that they risk fundamentally altering the strategic character and delivery of some of the BBC's services in Wales.
"We are also concerned that the proposals risk creating a position where the BBC could potentially be in breach of the public purposes to which it has committed to uphold in the Royal Charter."
The Royal Charter is a document which sets out the BBC's objectives and governance and is renegotiated every ten years.
Matt Brittin said Wales is "central" to the BBC's vision
In a reply, dated 2 November, Brittin responded that "the BBC takes its obligations to Wales extremely seriously".
"Wales is central to the BBC's mission as a public service broadcaster and to our role as a UK-wide institution," he said.
"As our response to the government's Charter Review green paper demonstrates, we believe the BBC must continue to serve every nation of the United Kingdom strongly, reflecting their democratic, cultural and linguistic life while ensuring that audiences see themselves represented across our services.
"Wales is fundamental to that vision.
"BBC Cymru Wales is one of Wales' most important national institutions."
The first minister's letter also raises worries about funding for S4C and on radio services said: "In the case of BBC Radio Cymru, this is the only radio news service through the medium of Welsh.
"The loss of services on BBC Radio Cymru doesn't just limit plurality; it removes services altogether.
"In Wales, the BBC has a unique role and unique responsibility in supporting the Welsh language. The first round of proposed savings identified raised a number of serious concerns.
"For BBC Radio Cymru, the provision of news, entertainment and culture will be significantly diminished by the current proposals.
"The same is true of BBC Radio Wales, with, as we understand it, an equally significant reduction in the number of feature programmes, which will compromise the breadth of the programme offer."
Brittin replied: "Welsh-language services are not peripheral to the BBC's public purposes; they are a core part of how we serve audiences in Wales.
"We fully recognise the unique role the BBC plays in supporting and promoting the Welsh language and we remain committed to ensuring that Welsh-speaking audiences continue to be well served through our services and partnerships."
Brittin is due to meet ap Iorwerth and Fychan on 18 November.
Rhun ap Iorwerth is a former BBC journalist and was first elected as a Senedd Member in 2013.
Rhuanedd Richards is a former chief executive of Plaid Cymru.
