ABSTRACT The origin of arcmin-sized odd radio circles (ORCs) found in modern all-sky radio surveys remains uncertain, with explanations ranging from starburst/active galactic nucleus-driven shocks to supernova remnants (SNRs) in the low-density ambient medium. Using well-calibrated radio light-curve models, we assess the possibility that ORCs are radio SNRs in low ambient densities. Our models imply that if ORCs 1–5 are SNRs, they must be within 200–350 kpc from the Sun, given their observed flux densities and sizes. To be evolving in the circumgalactic medium of the Milky Way, our models imply ORCs 1–5 to be ejecta-dominated SNRs within 50 kpc, evolving in ambient densities of (0.2–1.2) × 10−3 cm−3. However, this is statistically unlikely because ORCs 1–5 would have ages <640 yr, much smaller than their expected lifetimes of ≳105 yr at these densities. Additionally, the low SN rate implies only a few SNRs within 50 kpc. On the other hand, the circumgalactic medium SNR scenario for J0624−6948 is more likely (although still low probability) compared to ORCs 1–5, as our models allow J0624−6948 to be ≲3000 yr. The interpretation of J0624−6948 as a Sedov–Taylor SNR in the Large Magellanic Cloud is also possible for a wide range of ambient densities (6 × 10−4 to 0.5 cm−3), consistent with the local H i environment, and ages ∼(0.2–2.6) × 104 yr. Our work implies that while some ORCs may be SNRs, others are more likely large-scale shocks in distant galaxies.