Numerical Study with detailed chemistry has been conducted to investigate the flame structure and NOx formation characteristics in oxygen -enhanced(CH$_4$/O$_2$-$N_2$) and oxygen-enhanced-EGR(CH$_4$/O$_2$-$CO_2$) counter diffusion flame with various strain rates. A small amount of $N_2$is included in oxygen-enhanced-EGR combustion, in order to consider the inevitable $N_2$contamination by $O_2$production process or air infiltration. The results are as follows : In CH$_4$/O$_2$-$CO_2$flame it is very important to adopt a radiation effect precisely because the effect of radiation changes flame structure significantly. In CH$_4$/O$_2$-$N_2$flame special strategy to minimize NO emission is needed because it is very sensitive to a small amount of $N_2$. Special attention is needed on CO emission by flame quenching, because of increased CO concentration. Spatial NO production rate of oxygen-enhanced combustion is different from that of air and oxygen-enhanced-EGR combustion in that thermal mechanism plays a role of destruction as well as production. In case CH$_4$/O$_2$-$CO_2$flame contains more than 40% $CO_2$it is possible to maintain the same EINO as that of CH$_4$/Air flame with accomplishing higher temperature than that of CH$_4$/Air flame. EINO decreases with increasing strain rate, and those effects are augmented in CH$_4$/O$_2$flame.