A significant fraction (estimated to be as large as 30%) of the carbon monoxide in the atmosphere is produced by the oxidation of methane. One of the approaches that has been used to determine this fraction is an inversion of observed carbon and oxygen isotopic abundance ratios, C-13/C-12 and O-18/O-16, together with estimates of the magnitudes of other sources of carbon monoxide, and isotope effects in each source. For this purpose, values of the kinetic isotope effects in the methane oxidation are required, and although these have been determined experimentally for C-13, they have not been measured for O-18. This article examines the kinetic mechanism of methane oxidation and shows that an oxygen isotope effect could take place in the addition reaction of methyl radicals with molecular oxygen to form methylperoxy. Subsequent reactions in the complicated mechanism for methane oxidation do not lead to isotopic fractionation of oxygen. The expected kinetic isotope effect is calculated using variational transition state theory for the dissociation of methylperoxy, and combining the rate coefficient ratio with the ratio of equilibrium constants to obtain the ratio of recombination rate coefficients. The calculated kinetic isotope effects enrich O-18 in the methylperoxy adduct relative to molecular oxygen. The atmospheric implications of this are briefly discussed.