Moore swabs have been used extensively for passive sampling in wastewater surveillance, typically yielding presence/absence information for targets of interest. Quantitative analysis of Moore swab data is only possible if target uptake is well characterized, specifically the relationship between quantity of the target in the liquid sample matrix and the quantity of target sorbing to the Moore swab as a function of time. The mechanism of Moore swab absorption remains unclear and is important to understand toward using them more quantitatively. We conducted viral adsorption and desorption experiments using nonpathogenic SARS-CoV-2 surrogates: 6, MHV, and BCoV as well as heat-inactivated Zika virus (ZIKV). We fit empirical adsorption data from batch experiments to Langmuir, Freundlich and Redlich-Peterson isotherm models. We observed the adsorption behavior of viral targets onto Moore swabs is best characterized by the Redlich-Peterson isotherm model. Moore swabs retained the highest viral RNA concentrations after exposure durations between 9-12 hours in the presence of target microbes during kinetic viral adsorption experiments. The results inform current and future use of Moore swabs to produce quantitative data during wastewater surveillance, especially in settings where composite sampling remains infeasible.