The purpose of present paper is to extend the study of λ-Bernstein operators introduce by Cai et al. (J Inequal Appl 12:1–11, 2018). In our paper we consider a generalization of the Unρ operators introduced in 2007 by Radu Paltanea, using the new Bernstein–Bézier bases { b~ n,k} with shape parameter λ. Some approximation properties are given, including local approximation, error estimation in terms of moduli of continuity and Voronovskaja-type asymptotic formulas. Finally, we give some numerical examples and graphs to put in evidence the convergence of Unρ(f;x) to f(x).