A detailed analysis is presented for a boundary value problem of circular coils with parallel axes shielded by a cuboid of high permeability. Field solutions are given by establishing a suitable ansatz of the magnetic scalar potential, which can satisfy the boundary conditions on six surfaces of the cuboid without difficulty. Analytic expressions are also given for the self and mutual inductance of shielded circular coils with rectangular cross section. By differentiating the self- and mutual magnetic energy with respect to the centre coordinates of the shielded coils, the total forces exerted on them are further obtained, which consist of self and mutual force components. Finally, the numerical results of the proposed method are compared with those of the finite-element method simulations, and the proposed method proves to be accurate and efficient enough for practical applications.