Benchmark stars are crucial as validating standards for current as well as future large stellar surveys of the Milky Way. However, the number of suitable metal-poor benchmark stars is currently limited, owing to the difficulty in determining reliable effective temperatures (Teff) in this regime. We aim to construct a new set of metal-poor benchmark stars, based on reliable interferometric effective temperature determinations and a homogeneous analysis. The aim is to reach a precision of 1% in Teff, as is crucial for sufficiently accurate determinations of the full set of fundamental parameters and abundances for the survey sources. We observed ten late type metal-poor dwarf and giants: HD 2665, HD 6755, HD 6833, HD 103095, HD 122563, HD 127243, HD 140283, HD 175305, HD 221170 and HD 224930. Only three of the ten stars (HD 103095, HD 122563 and HD 140283) have previously been used as benchmark stars. For the observations, we used the high angular resolution optical interferometric instrument PAVO at the CHARA array. We modelled angular diameters using 3D limb darkening models and determined effective temperatures directly from the Stefan-Boltzmann relation, with an iterative procedure to interpolate over tables of bolometric corrections. Surface gravities (log(g)) were estimated from comparisons to Dartmouth stellar evolution model tracks. We collected spectroscopic observations from the ELODIE and FIES spectrographs and estimated metallicities ([Fe/H]) from a 1D non-LTE abundance analysis of unblended lines of neutral and singly ionized iron. We inferred Teff to better than 1% for five of the stars (HD 103095, HD 122563, HD 127243, HD 140283 and HD 224930). The effective temperatures of the other five stars are reliable to between 2-3%; the higher uncertainty on the Teff for those stars is mainly due to their having a larger uncertainty in the bolometric fluxes. We also determined log(g) and [Fe/H] with median uncertainties of 0.03dex and 0.09dex, respectively. This study presents reliable and homogeneous fundamental stellar parameters for ten metal-poor stars that can be adopted as a new set of benchmarks. The parameters are based on our consistent approach of combining interferometric observations, 3D limb darkening modelling and spectroscopic observations. The next paper in this series will extend this approach to dwarfs and giants in the metal-rich regime.