Concerns of possible transmission of SARS-CoV-2 from donors to patients by corneal transplantation have caused a decline in corneal transplantations. Graft culture media are routinely tested for infectious risks, but it is unclear whether this constitutes a viable means to avoid transmitting SARS-CoV-2 via keratoplasty. We found that SARS-CoV-2 RNA was not present in the medium after seven days of organ culture of corneas from donors (n = 4), who were SARS-CoV-2-positive upon tissue procurement. These medium samples showed no presence of viral RNA. To pursue this question under controlled conditions and further exclude the possibility of productive infection in corneal grafts, we inoculated corneoscleral discs from healthy donors (n = 8) with infectious SARS-CoV-2 and performed PCR testing of the culture medium at various time points. After seven days of culture, we also tested for SARS-CoV-2 RNA within the inoculated corneal tissue. The medium from tissue samples inoculated with SARS-CoV-2 showed no increase in viral RNA, which may indicate lack of viral replication in these corneal grafts. SARS-CoV-2-RNA was, however, detected on or in corneal tissue seven days after inoculation. Our data suggest that corneal grafts may not be permissive for replication of SARS-CoV-2 and demonstrates that PCR testing of culture media cannot safely exclude that tissue has been exposed to SARS-CoV-2. It also demonstrates the difficulty to differentiate between virus adherence and virus replication by PCR testing in SARS-CoV-2 exposed tissue.