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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≳ 10^5.3M_⊙). 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.