We derive the 3D-NLTE lithium abundance in the solar photosphere from the Lii line at 670 nm as measured in several solar atlases. The Li abundance is obtained from line profile fitting with 1D/3D-LTE/3D-NLTE synthetic spectra, considering several possibilities for the atomic parameters of the lines blending the Li feature. The 670 nm spectral region shows considerable differences in the two available disc-centre solar atlases, while the two integrated disc spectra are very similar. We obtain A(Li)3D–NLTE = 1.03. The 1D-LTE abundance is 0.07 dex smaller. The line-lists giving the best fit for the Sun may fail for other stars, while some line-lists fail to reproduce the solar profile satisfactorily. We need a better knowledge of the atomic parameters of the lines blending the Li feature in order to be able to reproduce both the solar spectrum and the spectra of other stars. An improved line-list is also required to derive reliable estimates of the isotopic Li ratio in solar-metallicity stars.