In this work, we prove that the set of link-homotopy classes of generalized string links over a closed, connected and orientable surface M of genus g ≥ 1 form a group, denoted by B_n(M) and we find a presentation for it. Moreover, we prove that its normal subgroup PB_nn(M), namely, the homotopy string links over M, is bi-orderable. These results extend results proved by Juan GonzalezMeneses in [GM], [GM2] and Ekaterina Yurasovskaya in [Y], respectively. Also, we obtain an exact sequence for link-homotopy braid groups, which is an extension of [Go, Theorem 1].

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