This paper is devoted to the study of a nonsymmetric Keyfitz-Kranzer system of conservation laws with the generalized and modified Chaplygin gas pressure law, which may admit delta shock waves, a topic of interest. Firstly, we solve the Riemann problems with piecewise constant data having a single discontinuity. For the generalized Chaplygin gas pressure law, the solution consists of three different structures: R + J , S + J , and δ . Existence and uniqueness of delta shock solution are established under the generalized Rankine-Hugoniot relation and entropy condition. For the modified Chaplygin gas pressure law, the structures of solution are R + J and S + J . Secondly, we discuss the limits of Riemann solutions for the modified Chaplygin gas pressure law as the pressure law tends to the generalized Chaplygin gas one. In particular, for some cases, the solution S + J tends to a delta shock wave, and it is different from the delta shock wave for the generalized Chaplygin gas pressure law with the same initial data. Thirdly, we simulate the Riemann solutions and examine the formation process of delta shock wave by employing the Nessyahu-Tadmor scheme. The numerical results are coincident with the theoretical analysis.