Background The literature abounds with increasing numbers of research studies using genomic data in combination with health data (eg, health records and phenotypic and lifestyle data), with great potential for large-scale research and precision medicine. However, concerns have been raised about social acceptability and risks posed for individuals and their kin. Although there has been public engagement on various aspects of this topic, there is a lack of information about public views on data access models. Objective This study aimed to address the lack of information on the social acceptability of access models for reusing genomic data collected for research in conjunction with health data. Models considered were open web-based access, released externally to researchers, and access within a data safe haven. Methods Views were ascertained using a series of 8 public workshops (N=116). The workshops included an explanation of benefits and risks in using genomic data with health data, a facilitated discussion, and an exit questionnaire. The resulting quantitative data were analyzed using descriptive and inferential statistics, and the qualitative data were analyzed for emerging themes. Results Respondents placed a high value on the reuse of genomic data but raised concerns including data misuse, information governance, and discrimination. They showed a preference for giving consent and use of data within a safe haven over external release or open access. Perceived risks with open access included data being used by unscrupulous parties, with external release included data security, and with safe havens included the need for robust safeguards. Conclusions This is the first known study exploring public views of access models for reusing anonymized genomic and health data in research. It indicated that people are generally amenable but prefer data safe havens because of perceived sensitivities. We recommend that public views be incorporated into guidance on models for the reuse of genomic and health data.