In the present paper, we study a new type of Bernstein operators depending on the parameter lambda is an element of [-1, 1]. The Kantorovich modification of these sequences of linear positive operators will be considered. A quantitative Voronovskaja type theorem by means of Ditzian-Totik modulus of smoothness is proved. Also, a Gruss-Voronovskaja type theorem for lambda-Kantorovich operators is provided. Some numerical examples which show the relevance of the results are given.