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The paper presents an approximate analytical solution in closed form for the mode of a weakly guiding circular optical fiber with an arbitrary profile of the refractive index. The solution is valid for guided linearly polarized modes of arbitrary radial and azimuthal orders. The proposed solution is based on the joint application of the Gaussian approximation method and the stratification method, as well as the replacement of an optical fiber with a bending radius significantly exceeding the radius of the fiber, some direct optical fiber with an equivalent profile of the refractive index. The results of calculations for a typical refractive index profile of a multimode optical fiber with a core and cladding diameters of 50/125 are presented. It is shown that for the considered examples, the degree of influence of the bending radius on the propagation constant of the mode of a curved fiber waveguide increases with increasing mode order. And, in the range of changes in the bend radius of the fiber, in which the conditions for applying the proposed approximate analytical solution are fulfilled, the mode propagation constant due to the bending of the fiber changes insignificantly and we can assume that this change does not depend on the value of the bend radius.