Influenza A virus (IAV) is a negative-sense RNA virus that causes seasonal infections and periodic pandemics, inflicting huge economic and human costs on society. The current production of influenza virus for vaccines is initiated by generating a seed virus through the transfection of multiple plasmids in HEK293 cells followed by the infection of seed viruses into embryonated chicken eggs or cultured mammalian cells. We took a system design and synthetic biology approach to engineer cell lines that can be induced to produce all viral components except hemagglutinin (HA) and neuraminidase (NA), which are the antigens that specify the variants of IAV. Upon the transfection of HA and NA, the cell line can produce infectious IAV particles. RNA-Seq transcriptome analysis revealed inefficient synthesis of viral RNA and upregulated expression of genes involved in host response to viral infection as potential limiting factors and offered possible targets for enhancing the productivity of the synthetic cell line. Overall, we showed for the first time that it was possible to create packaging cell lines for the production of a cytopathic negative-sense RNA virus. The approach allows for the exploitation of altered kinetics of the synthesis of viral components and offers a new method for manufacturing viral vaccines.