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Many approaches for the design of observers were developed to estimate the state of a linear time-delay system. This paper deals with the problem of observer synthesis for a class of two-dimensional discrete systems with delays. The results obtained are based on the exploitation of linear matrix inequality (LMI) techniques in combination with the Lyapunov equations. The design of twodimensional state observers for two-dimensional discrete-time systems with delays described by the second Fornasini-Marchesini model is formulated using new conditions in terms of linear matrix inequalities in order to facilitate the research parameters of the obtained observer. The efficiency and convergence of the proposed method are demonstrated by a numerical example.