We proposed a new family of noncoaxial Gauss-truncated Bessel beams through multiplying conventional symmetrical Bessel beams by a noncoaxial Gauss function. These beams can also be regarded as the exponential-truncated version of Bessel-Gauss beams since they can be transformed into the product of Bessel-Gauss beams and an exponential window function along a certain Cartesian axis. The closed-form solutions of the angular spectra and paraxial propagation of these beams were derived. These beams have asymmetrical intensity distributions and carry the same orbit angular momentum per photon as the corresponding Bessel-Gauss beams. While propagating along the z axis, the mth (m≠0) noncoaxial Bessel-Gauss beams rotate their intensity distributions and the mth-order vortex at the beam center has a transverse shift along the direction perpendicular to the offset axis. Depending on the product of the transverse scalar factor of the Bessel beams and the offset between the Gaussian window function and the center of the Bessel beams, the noncoaxial Bessel-Gauss beams can produce unit vortices with opposite signs in pairs during propagation.