We investigate the role of kinetic instabilities driven by a proton anisotropy on the onset of magnetic reconnection by means of two-dimensional hybrid simulations. The collisionless tearing of a current sheet is studied in the presence of a proton temperature anisotropy in the surrounding plasma. Our results confirm that anisotropic protons within the current sheet region can significantly enhance/stabilize the tearing instability of the current. Moreover, fluctuations associated with linear instabilities excited by large proton temperature anisotropies can significantly influence the stability of the plasma and perturb the current sheets, triggering the tearing instability. We find that such a complex coupling leads to a faster tearing evolution in the $T_\perp >T_\Vert$ regime when an ion-cyclotron instability is generated by the anisotropic proton distribution functions. On the contrary, in the presence of the opposite anisotropy, fire-hose fluctuations excited by the unstable background protons with $T_\Vert <T_\perp$ are not able to efficiently destabilize current sheets, which remain stable for a long time after fire-hose saturation. We discuss possible influences of this novel coupling on the solar wind and heliospheric plasma dynamics.