A series of anionic homopolymers, poly(sodium 2-(acrylamido)-2-methyl-1-propanesulfonate) (PAMPS) and amphiphilic copolymers of AMPS and sodium 11-(acrylamido)undecanoate (AaU), both block (PAMPS75-b-PAaUn), and random (P(AMPSm-co-AaUn)), were synthesized and their antiviral activity against Zika virus (ZIKV) was evaluated. Interestingly, while the homopolymers showed limited antiviral activity, the copolymers are very efficient antivirals. This observation was explained considering that under the conditions relevant to the biological experiments (pH 7.4 PBS buffer) the macromolecules of these copolymers exist as negatively charged (zeta potential about −25 mV) nanoparticles (4–12 nm) due to their self-organization. They inhibit the ZIKV replication cycle by binding to the cell surface and thereby blocking virus attachment to host cells. Considering good solubility in aqueous media, low toxicity, and high selectivity index (SI) of the PAMPS-b-PAaU copolymers, they can be considered promising agents against ZIKV infections.