Background: In Oman, many extended families tend to live in one household. Some families can include 20–30 individuals with the majority of them being children. This study investigates the role of children in spreading SARS-CoV-2 causing COVID-19 within family clusters in Oman. Methods: This retrospective study includes data of 1026 SARS-CoV-2 positive children (≤18 years) collected from the national surveillance database for COVID-19 between 1 February 2020 and 30 May 2020. Results: We included 1026 patients. Most, 842 were Omani (82%), 52% male, and 28.5% asymptomatic. Close to the half of symptomatic 419 (40%), patients presented with fever associated with other respiratory symptoms. Fifty pediatric patients were index cases who transmitted the virus to 107 patients in total (86 adults and 21 children) with a mode of 1. There is no statistical significance of all studied risk factors in the transmission of the SARS-CoV-2 virus including age, gender, and cycle threshold (CT) value. Conclusions: According to this study, children are not to be considered a significant driver of transmission of SARS-CoV-2 in Oman.