Let X be a subvariety of the n-dimensional algebraic torus. We study the set of algebraic points of X which, in a certain sense, “lie close” to an algebraic subgroup of dimension m. If m dim X < n we show that the set of these points has bounded height after removing from X a purely geometrically defined Zariski closed subset. The boundedness of height extends a result of Bombieri, Masser, and Zannier in the case r=1 and of Zannier in the case r=n-1. We also discuss a finiteness result for the points described above which depends on a recent result of Amoroso and David on lower bounds of heights of subvarieties of the torus.