The diffuse interstellar bands (DIBs) at 9577 and 9632{AA} are attributed to electronic transitions of the C60+ fullerene ion; thus, its column density in molecular clouds may be inferred. A comparison with the column density of the neutral C60 fullerene makes it possible to determine local physical conditions in the clouds. We aim to detect absorption lines of neutral C60 in the visible spectra of stars toward interstellar clouds. We searched for the {gamma}0, A0, and A1 absorption bands of C60 near 6070{AA}, 4024{AA}, and 3980{AA} using the Potsdam Echelle Polarimetric and Spectroscopic Instrument at the Large Binocular Telescope. We measured spectra toward eight background stars including BD+31 643. None of the {gamma}0, A0, and A1 absorption bands of C60 were detected in our spectra. Toward BD+31 643, we determine an upper limit of (14+/-11)x10^12^cm^-2^ for the column density of C60, and values in the range of 10^12^-10^14^cm^-2^ toward other stars. Using literature data for C60+, we derive upper limits of 0.8-3.8 for the C60:C60+ column density ratios. The non-detection of the {gamma}0 absorption band toward BD+31 643 is in conflict with a measurement employing infrared (IR) emission bands of C60. The discrepancy may indicate that free C60 molecules are not the source of the observed IR emission bands toward the star and that the IR emission arises from C60 molecules in dust grains instead. Alternatively, the oscillator strength of the {gamma}0 band may not be well constrained from the laboratory measurements.