We construct realizations for the 2-toroidal Lie algebra associated with the Lie algebra A1 using vertex operators based on bosonic fields. In particular our construction realizes higher-level representations of the 2-toroidal algebra for any given pair of levels (k0, k1) with k0≠0. We also construct a smaller module of level (k0, 0) for the toroidal algebra from the Fock space using certain screening vertex operator, and this later representation generalizes the higher-level construction of the affine Lie algebra sl2.