Mr McAvoy said he would not have been able to afford to study elsewhere.
The open letter from actors, musicians and directors called on the government to intervene.
Mr McAvoy added: "I'm intensely proud to be Scottish but I'm even more proud to be product of a Scottish training and I just hope future generations can benefit from the same kind of fantastic world-class training which I received."
In his letter to Mr McAvoy the first minister said: "Despite some suggestions to the contrary, the Academy is not facing an immediate funding gap."
But he acknowledged the difficulties caused as a result "of the historical imbalance between the funding it receives for teaching music and drama."
He added:"It would be inappropriate for the Scottish Government or the Scottish Funding Council to intervene in the running of such a specialised institution."
The funding council will now undertake a review of the funding of drama provision across Scotland.
Labour MSP Pauline McNeill, whose constituency includes the RSAMD, had also urged the first minister to "take action on the issue before it is too late".
The announcement of the review has been welcomed by the RSAMD principal.
Professor John Wallace said: "While there are always financial pressures to be managed in any organisation, the academy is certainly in a stable financial position.
"The academy next week is to confirm staffing restructuring while retaining the quality learning experience it offers to our students."
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