Due to the interaction of the cells, cellular communications are carried out with emergent collective behaviors, such as synchronization. In this paper, a network of coupled genetic oscillators is studied by considering a two-dimensional framework and diffusive coupling. Each genetic oscillator is a simple gene element with delayed self-inhibitions, which can exhibit periodic and chaotic dynamics. The network is investigated under the variation of the diffusion coefficient, in both periodic and chaotic dynamics. The results indicate the emergence of the so-called chimera state, i.e., the coexistence of coherent and incoherent regions, for certain diffusion coefficients. The computing of the strength of incoherence as a statistical measure represents that the local dynamics of the elements have a significant role in the network behavior. Actually, in the chaotic dynamics, the chimera state emerges in higher diffusion coefficients and a wider range. The effect of the time delay on the network behavior is also investigated.