To avoid the poles and unreachable points, the barycentric rational Hermite interpolation (BRHI) can select the interpolation weights to acquire appropriate BRI functions. When approaching the interpolated function with horizontal asymptotes, the rational Hermite interpolation function is unable to keep the original horizontal asymptotes. In this paper, a barycentric rational Hermite interpolation method with horizontal asymptotes is proposed. Firstly, the condition of level preserving asymptotes is studied for BRHI. Secondly, an optimal weights optimization model is constructed by using the minimum Lebesgue constant as the objective function. The constraint conditions of optimization model of the BRHI include no pole, no inaccessible point, the interpolation function level preserving asymptotes, and the normalization of barycentric weights. Finally, the barycentric rational Hermite interpolation with horizontal-preserving asymptotes is obtained. The effectiveness of the algorithm is proved by numerical examples.