Let \(\mathcal H\) be a Hilbert space \({\mathcal D} \subseteq {\mathcal H}\), \(\eta \in (-1,1)\) and \(T: {\mathcal D} \rightarrow {\mathcal H}\) be given. The authors call \(T\) to be \(\eta\)-monotone if \(\langle Tx-Ty, x-y \rangle \geq \eta \parallel Tx-Ty \parallel \parallel x-y \parallel\) for all \(x,y \in {\mathcal D}\). This extends the notion of Minty-Browder monotonicity (\(T\) is called Minty-Browder monotone if \(\langle Tx-Ty, x-y \rangle \geq 0\) for all \(x,y \in {\mathcal D}\)). Several examples of \(\eta\)-monotone operators are constructed. The operator \(T-i_{{\mathcal D}}\) is proved to be \(\eta_T\)-monotone, where \(T\) is a Minty-Browder monotone operator, \({\mathcal D}\) is a convex open subset of \({\mathcal H}\), \(\eta_T\) is a certain constant defined in terms of the operator \(T\) and \(i_{{\mathcal D}}\) denotes the inclusion map from \(\mathcal D\) into \(\mathcal H\). The main result proves that a perturbation of a certain specific type of a Minty-Browder monotone operator yields a \(\eta\)-monotone operator for a suitably chosen \(\eta\).