Agrawal et al. (Boll. Unione Mat. Ital. 8:169-180, 2015) introduced a Stancu-type Kantorovich modification of the operators proposed by Ren and Zeng (Bull. Korean Math. Soc. 50(4):1145-1156, 2013) and studied a basic convergence theorem by using the Bohman-Korovokin criterion, the rate of convergence involving the modulus of continuity, and the Lipschitz function. The concern of this paper is to obtain Voronoskaja-type asymptotic result by calculating an estimate of fourth order central moment for these operators and discuss the rate of convergence for the bivariate case by using the complete and partial moduli of continuity and the degree of approximation by means of a Lipschitz-type function and the Peetre K-functional. Also, we consider the associated GBS (generalized Boolean sum) operators and estimate the rate of convergence for these operators with the help of a mixed modulus of smoothness. Furthermore, we show the rate of convergence of these operators (univariate case) to certain functions with the help of the illustrations using Maple algorithms and in the bivariate case, the rate of convergence of these operators is compared with the associated GBS operators by illustrative graphics.