MEGARA (Multi Espectrógrafo en GTC de Alta Resolución para Astronomía) is an optical (3650-9750 Å), fibre-fed, medium-high spectral resolution (R = 6000, 12 000 and 20 000) instrument for the Gran Telescopio CANARIAS (GTC) 10.4-m telescope, commissioned in the summer of 2017, and currently in operation. The scientific exploitation of MEGARA requires a stellar spectra library to interpret galaxy data and to estimate the contribution of the stellar populations. In this paper, we introduce the MEGARA-GTC spectral library, detailing the rationale behind the building of this catalogue. We present the spectra of 97 stars (21 individual stars and 56 members of the globular cluster M15, which are both subsamples taken during the commissioning runs, and 20 stars from our ongoing GTC Open-Time programme). The spectra have R = 20 000 in the HR-R and HR-I set-ups, centred at 6563 and 8633 Å, respectively. We describe the procedures to reduce and analyse the data. Then, we determine the best-fitting theoretical models to each spectrum through a χ2 minimization technique, to derive the stellar physical parameters, and we discuss the results. We have also measured some absorption lines and indices. Finally, we introduce our project to complete the library and the data base in order to make the spectra available to the community.