The ESA astrometric mission Gaia, due for launch in late 2011, will observe a very large number of asteroids (~ 350,000 down to the magnitude 20), most from the main belt, with an unprecedented positional precision (at the sub-milliarcsecond level). Such high-precision astrometry will enable to considerably improve the orbits of a large number of objects, and also to determine the masses of the largest asteroids by analyzing their gravitational pull during close encounters with smaller ones. A global solution involving simultaneously all the perturbers and the smaller targets should yield about hundred masses with a precision better than 30 percent. The knowledge of these masses will be a rich source of information on the physics of main-belt asteroids and will increase the accuracy of modern solar system ephemerides. We outline the principle of the mathematical method based on variational equations developed to solve for the orbital parameters and the masses of the largest bodies. Calculations have been performed by taking into account realistic simulation of the Gaia observations such as geometry, time sequence, magnitude and by considering possible close approaches among 350,000 asteroids. We give a list of asteroids whose mass can be estimated along with its formal precision.