We introduce and investigate the concept of two-Lipschitz operator ideal between pointed metric spaces and Banach spaces. We show the basics of this new theory and we give a procedure to create a two-Lipschitz operator ideal from a linear operator ideal. We apply our result to the ideals of strongly p-summing and compact linear operator to obtain their corresponding two-Lipschitz operator ideal. Also, we establish a natural relation between two-Lipschitz and bilinear maps and show that the two-Lipschitz factorable p-dominated operators are those which are associated to the well-known p-semi-integral bilinear operators