Council officials have also cast doubt on the policy, arguing current funding for the costly initiative is insufficient.
The warnings come two years after the Government announced plans to teach all primary pupils at least two modern languages in addition to their mother tongue – known as the 1+2 model.
Since then the expert Languages Working Group has recommended 35 improvements, including better training for teachers and greater support for pupils in the classroom.
The Scottish Government has already announced £120,000 to fund pilot projects to demonstrate ways in which Scottish schools can begin to move towards the 1+2 model of modern languages provision.
The Government also intends to provide an additional £4 million in the new financial year to support the development of local authority language plans.
However, in a written submission to the Scottish Parliament's European Committee, which is scrutinising the strategy, the Scottish Parent Teacher Council (SPTC) casts serious doubts over the policy.
The submission states: "While we agree with the principle of widening the experience of languages for primary school pupils, we have grave concerns that the resource implications have not been considered sufficiently. This will require teachers in primary school to hold appropriate qualifications in language teaching. This is currently not the case and will require a substantial investment and significant time to make this a reality."
It adds: "While we support aspirational approaches, we are concerned that 1+2 is simply an unachievable target within the foreseeable future and feel an initial target of 1+1 may be more realistic."
The SPTC said it was concerned there was insufficient resource available for the project and warned teachers did not currently have sufficient skills to make it work.
Cosla, which represents councils, said in its submission it was aware there was a need for "at least double or triple the previous language funding to assist local authorities" in taking forward the project.
Ades, which represents local authority heads of education, also raised concerns over the issue of funding.
Its submission said: "There is, quite clearly, insufficient funding available at this time to deliver the 35 recommendations of the report of the Languages Working Group.
"At present, only a minority of primary teachers in Scotland are equipped to deliver a modern languages curriculum.
"While undoubtedly there will be some teaching assistants who are able to support the teaching of a modern language, the current number of such staff across Scotland is likely to be very small."
The concerns come after a historic decline in the number of pupils taking modern language Higher exams.
The demise of languages in secondary has been blamed on the fact many schools no longer see languages as compulsory, despite school inspectors calling for them to be a "core element" in the first three years of secondary. In addition, as part of cuts to education budgets, two-thirds of local authorities have scrapped foreign language assistants.
There have also been problems in primary, with The Herald revealing three-quarters of schools were missing recommended targets for the delivery of modern languages.
The Scottish Government commitment mirrors the so-called Barcelona Agreement by the European Council which called for the teaching of at least two foreign languages from an early age.