The board's chairwoman Meri Huws said it believed the county council should appoint a bilingual person to the pound;100,000-a-year post of education director.
Geraint Davies, secretary of the National Association of Schoolmasters Union of Women Teachers Cymru, this week condemned the council's stance.
"Carmarthenshire has the largest percentage of Welsh speakers in Wales, and for the council to consider appointing a non-Welsh speaker is an insult to its people and to the language," he said.
"Carmarthenshire has a Welsh-medium education system that is regarded as among the best in the country, so in line with its own bilingual policy it needs to have a director who can communicate fluently through Welsh."
The council said when it first advertised the job - with Welsh described as "essential" - only five applications were received. It decided to re-advertise without the language requirement.
The row deepened when councillor Peter Hughes Griffiths, a member of the selection panel, refused to sit for interviews.
Mr Griffiths said: "I was highly surprised to be told at a panel meeting that there would be no assessment of linguistic skills.
"More than 100 schools in Carmarthenshire use Welsh as a first language."
He added that following a public meeting last week it was possible legal action would be taken to ensure a non-Welsh speaker did not land the post.
"Two of the county's past education directors spoke at the meeting and said that unless a person was bilingual the job would be impossible," he said.
UCAC, the Welsh-medium teachers' union, has written to the council expressing its dissatisfaction. General secretary Moelwen Gwyndaf said: "We were quite surprised that the ad did not include the ability to speak Welsh.
"If the director is unable to speak Welsh, then all his meetings with heads would have to be conducted in English and the whole ethos of the education system in Carmarthenshire would change."
Dismissing Mr Griffiths' remarks as "irresponsible", Mr Martin Morris, deputy leader of the county council, insisted that the authority merely wanted the best person for the job.
He said: "There is a danger that top-quality candidates could be put off from joining us."