Mysterious Sphere and Electrostatics

In the influence of the small charge, the charges of the the neutral sphere re-arrange themselves in such a way that they attract the small charge. This phenomena is known as image charge distribution.

The big mass will polarise itself, with negative charges concentrated near the smaller charged mass and opposite charges on the other side of the mass, though overall the bigger mass will still remain neutral owing to the same amount of positive and negative charges being on both the sides. Now the opposite charges on the bigger sphere will act on the smaller mass but the distance between them will play the vital role here. Negative charge being near to the smaller mass will attract it with the larger electrostatic force compared to the positive charge which will be induced to the farthest side of the bigger mass. so a net force of attraction will be applicable in the case.

Apart from this, nothing has been mentioned about the masses of the bodies. I believe a small force of attraction will be available due to the gravitational effect also.