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This note contains discussions on the entanglement entropy and mutual information of a strongly coupled field theory with a critical point which has a holographic dual. We investigate analytically, in the specific regimes of parameters, how these non-local operators behave near the critical point. Interestingly, we observe that although the mutual information is constant at the critical point, its slope shows a power-law divergence in the vicinity of the critical point. We show that the leading behavior of mutual information at and near the critical point could yield a set of critical exponents if we regard it as an order parameter. Our result for this set of static critical exponents is (1/2,1/2,1/2,2) which is identical to the one calculated via the thermodynamic quantities. Hence it suggests that beside the numerous merits of mutual information, this quantity also captures the critical behavior of the underlying field theory and it could be used as a proper measure to probe the phase structure associated with the strongly coupled systems., Comment: 25 pages, 2 figures