Multifunctional properties of matter, including dielectric and magnetic properties, are of great interest in physics and applications. Here, we investigate the dielectric properties and magnetoelectric effects in FeCo-MgF insulating nanogranular films. These films have a nanogranular structure in which Fe-Co alloy-based magnetic granules are dispersed in an Mg-F insulator-based matrix. Both a large dielectric constant of about 420 at 100-kHz and a large magnetoelectric effect of about 6.2% are obtained at room temperature. Insulating nanogranular films are a new class of multifunctional materials. The dielectric properties and the magnetoelectric effects originate in the nano-structure which consists of a magnetic metal and an insulator, and are explained by the spin-dependent tunneling charge oscillation between a pair of magnetic granules.