We compare a very deep X-ray image of M87, at the center of the Virgo Cluster, to high-quality optical images of the low excitation emission-line gas in the same region. There are striking coincidences of detail between the two. We explore the possiblity that this represents a thermal interaction between hot gas at 10^7 K and warm gas at 10^4 K. We find two temperatures are present in the X-ray gas, with the lower more prevelant in the vicinity of the optical filaments. Electron conduction from the hot phase to the cooler one provides a quantitatively acceptable energy source for the optical filaments, and we show additionally that it can do so for the brightest X-ray cluster, Perseus. If operative, conduction in the presence of gas-rich galaxy mergers, may explain the presence of "cool cores" in clusters of galaxies.

18 pages, 6 figures, 2 tables