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This paper initiates a general study of the spectrum of two-particle bound states of transfer matrices for a fairly wide class of Gibbs fields at high temperature T. In the present first part of this study, a detailed statement of the problem is given and the existence of a so-called “isolated level” lying at a distance ∼ 1/T2 from the boundary of the continuous spectrum is established for all values of the total quasimomentum Λ of the system. In the concluding part of the paper, we prove that there are no other bound states provided that Λ is far from certain singular values. In the second part, we will consider bound states for Λ close to the singular values. The distance from these states (adjacent levels, in the authors' terminology) to the continuous spectrum is at most of the order of 1/T4.