If the selfadjoint operator \(A\) on a Hilbert space \(H\) is such that \(mI\leq A\leq MI\), where \(0< m< M\), then the Kantorovich inequality says that \(1\leq\langle Ax, x\rangle\langle A^{-1} x,x\rangle\leq (m+ M)^2/(4mM)\) for any unit vector \(x\) in \(H\). In this paper, the author uses Grüss-type inequalities, which he obtained before, and their operator versions to establish inequalities of Kantorovich type for the more general class of operators \(A\) satisfying \(\text{Re}[(A^*- \overline\alpha I)(\beta I- A)]\geq 0\). The Grüss-type inequalities refer to the ones which give upper bounds for \(|\langle u,v\rangle-\langle u,e\rangle\langle e,v\rangle|\) for vectors \(u\), \(v\) and \(e\) in \(H\) with \(\| e\|= 1\) and scalars \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) satisfying \(\| u- ((\alpha+ \beta)/2)e\|\leq |\beta-\alpha|/2\) and \(\| v- ((\gamma+\delta)/2) e\|\leq |\delta- \gamma|/2\). There are also established estimates for \(\| A\|^2- w(A)^2\) and \(w(A)^2- w(A^2)\) for the above class of \(A\). Here, \(w(A)\) denotes the numerical radius \(\sup\{|\langle Ax, x\rangle|: x\in H\), \(\| x\|= 1\}\) of \(A\).