The governors say that they are writing in response to a previous letter signed by 28 teachers which, they allege, was critical of school inspectors and the acting-principal and vice-principal.
Despite the governors writing that "we do not expect it to see it on any website or other media," the letter has been obtained by the BBC.
An inspection by the Education and Training Inspectorate (ETI) published on 9 January highlighted "poor working relationships" at the 1,100 pupil post-primary school.
The governors' letter to all members of the teaching staff is also dated 9 January.
It alleges that "comments regarding the acting principal and vice principal" made in the previous letter were "totally unacceptable and potentially libellous".
It also said that there is "resistance to improvements by a small group of staff who consider they have no obligation or responsibility for the difficulties which the school has encountered".
The letter also criticises those teachers "with a post of responsibility in the management structure" who signed the earlier letter for failing to see their involvement in shaping the current situation.
It concludes by inviting teachers who feel "that they do not have the qualities or commitment necessary to professionally address the obvious deficiencies in the school" to "carefully consider their position".
The letter also mentions that the "school is subject to monitoring by Council for the Curriculum, Examinations and Assessment (CCEA) because of malpractice in the operation of De La Salle as an examination centre in 2016".
There have been a number of difficulties at De La Salle over the past year including staff disputes and protests by some parents.
The board of governors of De La Salle college in west Belfast have sent a letter to all 76 teachers there inviting them to "consider their positions".