Let \(p_{n,k}(x):=e^{-nx}(nx)^k/k!\), and \(J_{n,k}(x):=\sum_{j=k}^\infty p_{n,j}(x)\), \(n,k\geq0\). The authors set \(Q^{(\alpha)}_{n,k}(x):=J^\alpha_{n,k}(x)-J^\alpha_{n,k+1}(x)\), \(n,k\geq0\), and introduce the following Durrmeyer variant of the Szász approximation operators on \([0,\infty)\), associated with a function \(f\in L[0,\infty)\). Namely, \[ M_{n,\alpha}(f,x):=n\sum_{k=0}^\infty Q^{(\alpha)}_{n,k}(x)\int_0^\infty p_{n,k}(t)f(t)\,dt,\quad n=0,1,2,\dots. \] If \(\alpha=1\), then the operators reduce to those defined by S. M. Mazhar and V. Totik [Acta Sci. Math. (Szeged), 49, 257–269 (1985; Zbl 0611.41013)]. Under the assumptions that \(f\) is of bounded variation in every finite subinterval, and satisfies a growth condition \(| f(x)| \leq Ke^{\beta x}\), the authors obtain pointwise estimates on how close \(M_{n,\alpha}(f,x)\) is to the average \((f(x+)+\alpha f(x-))/(\alpha+1)\).