We present some theoretical predictions concerning the amplitude and magnetic sensitivity of the linear-polarization signals produced by scattering processes in the hydrogen Lyα line of the solar transition region. To this end, we have calculated the atomic-level polarization (population imbalances and quantum coherences) induced by anisotropic radiation pumping in semiempirical and hydrodynamical models of the solar atmosphere, taking into account radiative transfer and the Hanle effect caused by the presence of organized and random magnetic fields. The line-center amplitudes of the emergent linear-polarization signals are found to vary typically between 0.1% and 1%, depending on the scattering geometry and the strength and orientation of the magnetic field. The results shown here encourage the development of UV polarimeters for sounding rockets and space telescopes with the aim of opening up a diagnostic window for magnetic field measurements in the upper chromosphere and transition region of the Sun.