The amount of data in this world is getting higher, and overwriting technology also has severe challenges. Data growth is expected to grow to 175 ZB by 2025. Data storage technology in DNA is an alternative technology with potential in information storage, mainly digital data. One of the stages of storing information on DNA is synthesis. This synthesis process costs very high, so it is necessary to integrate compression techniques for digital data to minimize the costs incurred. One of the models used in compression techniques is the generative model. This paper aims to see if compression using this generative model allows it to be integrated into data storage methods on DNA. To this end, we have conducted a Systematic Literature Review using the PRISMA method in selecting papers. We took the source of the papers from four leading databases and other additional databases. Out of 2440 papers, we finally decided on 34 primary papers for detailed analysis. This systematic literature review (SLR) presents and categorizes based on research questions, namely discussing machine learning methods applied in DNA storage, identifying compression techniques for DNA storage, knowing the role of deep learning in the compression process for DNA storage, knowing how generative models are associated with deep learning, knowing how generative models are applied in the compression process, and knowing latent space can be formed. The study highlights open problems that need to be solved and provides an identified research direction.