Twisted bilayer graphene (tBLG) has recently emerged as a new platform for studying electroncorrelations, the strength of which can be controlled via the twist angle. Here, we study the effectof internal screening on electron-electron interactions in undoped tBLG. Using the random phaseapproximation, we find that the dielectric response of tBLG drastically increases near the magicangle and is highly twist-angle dependent. As a consequence of the abrupt change of the Fermivelocity as a function of wave vector, the screened interaction in real space exhibits attractiveregions for certain twist angles near the magic angle. Attractive interactions can induce chargedensity waves and superconductivity and therefore our findings could be relevant to understand themicroscopic origins of the recently observed strong correlation phenomena in undoped tBLG. Theresulting screened Hubbard parameters are strongly reduced and exhibit a non-linear dependence onthe twist angle. We also carry out calculations with the constrained random phase approximationand parametrize a twist-angle dependent Keldysh model for the resulting effective interaction.