By the use of a linear stability analysis, the analytic expression for the growth rate of the disperse modulational instability in the zigzag optical waveguide array with nonlinear coupling is got, and the influence of the nonlinear coupling on the modulational instability region for different values of the next-nearest-neighbor coupling coefficient is analyzed. The results indicate that varying the value of the nonlinear coupling parameter can evidently affect the shape of the modulational instability area. According to the result from the modulational instability analysis, the existence conditions of bright discrete solitons are predicted. Furthermore, analytical solutions for the bright discrete solitons in the present zigzag optical waveguide array are obtained by the multi-scale method. What is more, the existence conditions of such bright discrete solitons are discussed, which are manifested to be in agreement with the modulational instability analysis.