We study a new class of so-called quasi-infinitely divisible laws, which is a wide natural extension of the well known class of infinitely divisible laws through the Lévy--Khinchine type representations. We are interested in criteria of weak convergence within this class. Under rather natural assumptions, we state assertions, which connect a weak convergence of quasi-infinitely divisible distribution functions with one special type of convergence of their Lévy--Khinchine spectral functions. The latter convergence is not equivalent to the weak convergence. So we complement known results by Lindner, Pan, and Sato (2018) in this field.