The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine Bernstein-Durrmeyer variants. These new operators depend on a certain function τ defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the Ditzian-Totik modulus of smoothness are studied. Quantitative Voronovskaja–type theorems and Grüss-Voronovskaja–type theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given.