The residual internal stresses in a cylindrical wire produced in the rotating-water melt spinning process and a coated wire obtained by drawing from a melt have been calculated within the thermal viscoelasticity and structural relaxation theories. The coated wire consists of the core and the sheath with different thermal properties. The problem is considered with allowance made for the generation and the relaxation of stresses in the core and the sheath in the temperature range from initial (corresponding to the liquid state of a two-layer wire) to room temperature. The distributions of the residual stresses have been calculated for the free amorphous metallic wire and the amorphous wire with the sheath having a different elastic modulus and thermal expansion coefficient. The influence of preparation conditions and thermal properties of materials on the calculated parameters is analyzed.