We measure the molecular gas environment near recent (<100 yr old) supernovae (SNe) using ∼1″ or ≤150 pc resolution CO (2-1) maps from the PHANGS-Atacama Large Millimeter/submillimeter Array (ALMA) survey of nearby star-forming galaxies. This is arguably the first such study to approach the scales of individual massive molecular clouds (M mol ≳ 105.3 M ⊙). Using the Open Supernova Catalog, we identify 63 SNe within the PHANGS-ALMA footprint. We detect CO (2-1) emission near ∼60% of the sample at 150 pc resolution, compared to ∼35% of map pixels with CO (2-1) emission, and up to ∼95% of the SNe at 1 kpc resolution, compared to ∼80% of map pixels with CO (2-1) emission. We expect the ∼60% of SNe within the same 150 pc beam, as a giant molecular cloud will likely interact with these clouds in the future, consistent with the observation of widespread SN-molecular gas interaction in the Milky Way, while the other ∼40% of SNe without strong CO (2-1) detections will deposit their energy in the diffuse interstellar medium, perhaps helping drive large-scale turbulence or galactic outflows. Broken down by type, we detect CO (2-1) emission at the sites of ∼85% of our 9 stripped-envelope SNe (SESNe), ∼40% of our 34 Type II SNe, and ∼35% of our 13 Type Ia SNe, indicating that SESNe are most closely associated with the brightest CO (2-1) emitting regions in our sample. Our results confirm that SN explosions are not restricted to only the densest gas, and instead exert feedback across a wide range of molecular gas densities.