We here report a detailed high-pressure infrared transmission study of BiTeCl and BiTeBr. We follow the evolution of two band transitions: the optical excitation $beta$ between two Rashba-split conduction bands, and the absorption $gamma$ across the band gap. In the low pressure range, $p< 4$~GPa, for both compounds $beta$ is approximately constant with pressure and $gamma$ decreases, in agreement with band structure calculations. In BiTeCl, a clear pressure-induced phase transition at 6~GPa leads to a different ground state. For BiTeBr, the pressure evolution is more subtle, and we discuss the possibility of closing and reopening of the band gap. Our data is consistent with a Weyl phase in BiTeBr at 5$-$6~GPa, followed by the onset of a structural phase transition at 7~GPa.
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