The Bessel beams are often used for investigating the scattering properties of the non-diffracting beams, but the Bessel beams cannot be realized physically. As the pseudo-nondiffracting beams and the exact solution to the paraxial Helmholz equation, the Bessel-Gauss beams can be generated directly from the laser resonator and possess finite spatial width. The dimensionless scattering function is derived for the Bessel-Gauss beams scattered by a sphere by means of Fourier-transform, the plane wave expansion and the vector spherical wave expansion. By numerical simulation and comparison with that of the Bessel beam and the Gauss beam, it can be found that only the scattering intensity is influenced as the spherical scatterer is shifted off the beam axis, the directions where the scattering extreme points exist almost keep no changes. The scattering is dominant in the direction of the conical angle for the Bessel-Gauss beam and the Bessel beam, but for the Gauss beam the forward scattering is always dominant.