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Acoustic Bessel beams are commonly used as ideal sources to study the characteristics for acoustic-vortex (AV) beams, exhibiting prosperous perspectives in contactless object manipulations and acoustic communications. However, accurate Bessel beams are difficult to construct using two-dimensional arrays in practical applications. By integrating active phase control and passive phase modulation to a ring-array of sectorial planar transducers, quasi-Bessel acoustic-vortex (QB-AV) beams of arbitrary order are built by the line-focus of AV fields in the current study. Based on the Snell’s refraction law, a circular saw-tooth lens of phase modulation is designed to converge incident waves toward the beam axis at a same deflection angle. QB-AV beams constructed by the main lobes of the sectorial sources are demonstrated by theoretical derivations, numerical simulations and quality evaluations, while those created by the side lobes are neglected to avoid the pressure fluctuations in the near field. Experimental measurements for AV beams of different orders coincide basically with the simulations, demonstrating that line-focused QB-AV beams can be generated along the beam axis across the pressure peak. With the increase of the topological charge, the peak-pressure of the beam decreases accordingly with a reduced effective axial range. The favorable results prove that, as a special kind of diffraction sources, the adjustable QB-AV beams may enable more important biomedical applications where Bessel beams are necessary, especially for the line-focused manipulation of biological and drug particles.