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Multiscale interactions between the tearing mode (TM) and ion temperature gradient (ITG) turbulence are studied numerically using a self-consistent gyrofluid model in slab geometry. It is found that the multiscale system goes through five distinct phases and is then saturated in a dynamic quasi-steady state. During the nonlinear evolution, the macroscale TM and the microscale ITG turbulence can mutually destabilize each other. On the one hand, the fluctuation level of the turbulence is greatly raised when the magnetic island grows beyond a threshold. The contributions of different scale fluctuations to heat conductivity are calculated. Although the macroscale long wavelength TM plays a dominant role in inducing heat transport in comparison with micro turbulence, the secondary harmonics of the TM have a considerable effect on causing heat pinch. On the other hand, the island growth is significantly enhanced through increasing the ITG as the island width increases above a critical value or the island propagating velocity is reduced below a critical value. The underlying mechanisms of the mutual destabilizations are identified. In addition, the generation of zonal flows and the associated turbulent transport in the multiscale interaction process are analysed in detail.