While there is no minimum qualification, it is important that you should be specifically suited for the job in question. While at a Hungarian company, 20% of the employees can be foreigners with no questions asked, employers rarely invoke this rule, and usually start the work permit application procedure by submitting a labor demand to the local Labor Office. During the application process that follows, it will be determined if the position advertised can be filled by a Hungarian employee or if it can be assigned to a third-country national who is a better fit based on their work permit application documents.
If your employer wants you for the position, there must be a reason for it. Since the Labor Office gives preference to local work force, it is the employer’s responsibility to check if there are locals who might match the job requirements – and it might not hurt to discuss this aspect of your employment as well to avoid rejection based on local competition. However, if you have unique skills relevant for the position, you have good chances; just make sure to include all relevant supporting documents in your Hungarian work permit application, whether they are diplomas or other documents.