Carbon-14 is produced in the atmosphere when neutrons from cosmic-ray air showers are captured by nitrogen nuclei. Atmospheric carbon-14 becomes trapped in air bubbles in polar ice as compacted snow (firn) transforms into ice. Carbon-14 is also produced in situ in ice grains by penetrating cosmic-ray neutrons and muons. Recent ice core measurements indicate that in the carbon monoxide (CO) phase, the carbon-14 is dominated by the in situ cosmogenic component at most ice coring sites. Thus, it should be possible to use ice-bound CO to reconstruct the historical flux of cosmic rays at Earth, without the transport and deposition uncertainties associated with beryllium-10 or the carbon cycle uncertainties affecting atmospheric carbon dioxide. The measurements will be sensitive to the cosmic-ray flux above the energy range most affected by solar modulation. We present estimates of the expected sensitivity of CO in ice cores to the historical flux of Galactic cosmic rays, based on recent studies of CO in polar ice.