The novel SARS-CoV-2 virus has very high infectivity, which allows it to spread rapidly around the world. Attempts at slowing the pandemic at this stage depend on the number and quality of diagnostic tests performed. We propose that the olfactory epithelium from the nasal cavity may be a more appropriate tissue for detection of SARS-CoV-2 virus at the earliest stages, prior to onset of symptoms or even in asymptomatic people, as compared to commonly used sputum or nasopharyngeal swabs. Here we emphasize that the nasal cavity olfactory epithelium is the likely site of enhanced binding of SARS-CoV-2. Multiple non-neuronal cell types present in the olfactory epithelium express two host receptors, ACE2 and TMPRSS2 proteases, that facilitate SARS-CoV-2 binding, replication, and accumulation. This may be the underlying mechanism for the recently reported cases of smell dysfunction in patients with COVID-19. Moreover, the possibility of subsequent brain infection should be considered which begins in olfactory neurons. In addition, we discuss the possibility that olfactory receptor neurons may initiate rapid immune responses at early stages of the disease. We emphasize the need to undertake research focused on additional aspects of SARS-CoV-2 actions in the nervous system, especially in the olfactory pathway.