Genome sequencing can be used to actively search for genetic variants unrelated to the initial clinical question. While such 'opportunistic genomic screening' (OGS) has been proposed in the USA, a European discussion on the ethics of OGS is only starting. Should testing for selected 'secondary findings' be offered to patients who need genetic sequencing? Using focus groups and interviews, we explored views on OGS in adults and minors from three perspectives: policy experts (n = 9), health professionals (n = 8) and patient representatives (n = 7). A thematic approach was used to analyze the data. There was consensus that OGS should be evaluated in terms of the classical 'screening' framework, rather than as a form of 'good patient care'. Accordingly, stakeholders agreed that professionals do not have a 'fiduciary duty' to look for secondary findings. Adding screening to clinical care was only conceivable with the patient's informed consent. In general, stakeholders were reluctant towards OGS. Arguments for regarding OGS being premature included lack of evidence regarding its clinical utility, also in view of uncertainties regarding general population penetrance, and concerns about both its psychosocial impact and respect for autonomy. All groups agreed that OGS means unequal access, which was seen as problematic. Yet, despite their concerns, stakeholders felt that offering screening for certain actionable pathogenic variants with known high penetrance could potentially be valuable in certain contexts for both adults and minors. Pharmacogenetic variants were regarded as a category by itself, for which OGS could potentially be beneficial.