Loss of smell (anosmia) and loss of taste (dysgeusia) have been commonly noted in people with COVID-19 infection, including those without any other symptoms such as fever, cough or breathing difficulty.
In a recent case, anosmia developed on day 7 of infection. MRI showed edema (swelling) and increased signal of olfactory bulbs and tracts. Patient recovered on day 24 and the repeat MRI showed normal findings.

Conclusion
1. Loss of smell and taste can occur in patients with COVID-19 infection, even in the absence of other symptoms.
2. Loss of smell is reversible and patients usually recover their sense of smell and taste within 4 weeks.
3. The underlying basis of anosmia seems to be involvement of olfactory nervous system (olfactory bulbs and tracts by the virus through trans-nasal spread.
(Source: Neurology, May 22, 2020)