In this paper, we study the effects of hydrogen enrichment and depletion on greenhouse gas emissions from a bluff-body stabilized turbulent flame fed with a mixture of methane and hydrogen as its fuel. To do this study, we use the chemical mechanism consisting of 70 species and 463 elementary reactions, adopt flamelet combustion model to surmount the closure problem of chemical source terms within the classical Reynolds-Stress-Model RSM turbulence approaches. We employ the two-equation standard κ-ε turbulence model with round jet corrections incorporated with suitable wall functions. The turbulence-chemistry interaction is taken into account using the presumed-shape probability density functions PDFs. The radiation effects of the most important radiating species are taken into account supposing an optically-thin flame. Using such detailed-chemistry approach, we first simulate a benchmark burner to evaluate the capability of our developed numerical simulation to predict the flame structure. Second, we compare the achieved distributions of temperature and species concentrations within the flame with the available measurement data. The achieved results indicate that our numerical simulation predicts the temperature and species concentration distributions very accurately and that the achieved results are in great agreement with the experimental data. Then, we investigate the effects of enriching fuel on the achieved results. We also compare the results with those of the benchmark case in terms of the greenhouse gas emissions right at the outlet of combustor. Our findings show that the greenhouse gas emissions can be reduced via using suitable hydrogen enrichment techniques. Alternatively, greenhouse gas emissions can increase with hydrogen fuel depletion.