Summary: In this somewhat didactic note we give a detailed alternative proof of the known result of C. Z. Wei and J. Winnicki [Stochastic Processes Appl. 31, No. 2, 261–282 (1989; Zbl 0673.60092)] which states that, under second-order moment assumptions on the offspring and immigration distributions, the sequence of appropriately scaled random step functions formed from a critical Galton-Watson process with immigration (not necessarily starting from zero) converges weakly towards a squared Bessel process. The proof of Wei and Winnicki [loc. cit.] is based on infinitesimal generators, while we use limit theorems for random step processes towards a diffusion process due to M. Ispány and G. Pap [Acta Math. Hung. 126, No. 4, 381–395 (2010; Zbl 1274.60109)]. This technique was already used by M. Ispány [Publ. Math. 72, No. 1–2, 17–34 (2008; Zbl 1164.60061)], who proved functional limit theorems for a sequence of some appropriately normalized nearly critical Galton-Watson processes with immigration starting from zero, where the offspring means tend to its critical value 1. As a special case of Theorem 2.1 of Ispány [2008, loc. cit.] one can get back the result of Wei and Winnicki [loc. cit.] in the case of zero initial value. In the present note we handle nonzero initial values with the technique used by Ispány [2008, loc. cit.], and further, we simplify some of the arguments in the proof of Theorem 2.1 of Ispány [2008, loc. cit.] as well.