Oxy-fuel combustion is one method to produce concentrated streams of carbon dioxide for subsequent sequestration. An additional benefit of oxy-firing is a reduction in NOx formation. The high combustion temperatures resulting from oxy-firing are typically controlled by exhaust gas recirculation. In this work, we performed chemical kinetic (CHEMKIN) calculations using a mechanism validated for these conditions to study the effects of dilution by either carbon dioxide or water vapor on methane oxy-combustion, and to compare the results with methane air-combustion (N-2 as the diluent). The study was performed under adiabatic conditions at a pressure of P = 30 atm, an equivalence ratio of phi = 1, and initial temperatures of T = 800-1200 K, which mimic the inlet conditions of many gas turbines and flameless combustors. The calculations show that H2O addition at low initial temperatures and high pressure leads to considerable reduction in the ignition delay time. This result is mainly due to changes in the radical pool and competition between the elementary reactions for the hydroxyl (OH), methyl-peroxyl (CH3OO), and methoxy (CH3O) radicals at low temperatures. On the other hand, carbon dioxide leads to lower adiabatic flame temperatures and flame speeds at elevated pressure. One reason for this effect of CO2 is its higher specific heat capacity and lower thermal diffusivity, compared to N-2 and H2O. In addition, carbon dioxide 'dilution decreases the rate of the main chain branching reaction (R29: H + O-2 <-> O + OH) due to increasing competition between this reaction and the reverse of (R29: CO + OH <-> CO2 + H) for H radicals, and thus results in reductions in flame propagation and flame speed.