Context. Ground-based high-resolution spectrographs are key instruments for several astrophysical domains, such as exoplanet studies. Unfortunately, the observed spectra are contaminated by the Earth’s atmosphere and its large molecular absorption bands. While different techniques (forward radiative transfer models, principle component analysis (PCA), or other empirical methods) exist to correct for telluric lines in exoplanet atmospheric studies, in radial velocity (RV) studies, telluric lines with an absorption depth of >2% are generally masked, which poses a problem for faint targets and M dwarfs as most of their RV content is present where telluric contamination is important. Aims. We propose a simple telluric model to be embedded in the Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO) data reduction software (DRS). The goal is to provide telluric-free spectra and enable RV measurements through the cross-correlation function technique (and others), including spectral ranges where telluric lines fall. Methods. The model is a line-by-line radiative transfer code that assumes a single atmospheric layer. We use the sky conditions and the physical properties of the lines from the HITRAN database to create the telluric spectrum. This high-resolution model is then convolved with the instrumental resolution and sampled to the instrumental wavelength grid. A subset of selected telluric lines is used to robustly fit the spectrum through a Levenberg-Marquardt minimization algorithm. Results. We computed the model to the H₂O lines in the spectral range of ESPRESSO. When applied to stellar spectra from A0- to M5-type stars, the residuals of the strongest water lines are below the 2% peak-to-valley (P2V) amplitude for all spectral types, with the exception of M dwarfs, which are within the pseudo-continuum. We then determined the RVs from the telluric-corrected ESPRESSO spectra of Tau Ceti and Proxima. We created telluric-free masks and compared the obtained RVs with the DRS RVs. In the case of Tau Ceti, we identified that micro-telluric lines introduce systematics up to an amplitude of 58 cm s⁻¹ and with a period of one year if not corrected. For Proxima, the impact of micro-telluric lines is negligible due to the low flux below 5900 A. For late-type stars, the gain in spectral content at redder wavelengths is equivalent to a gain of 25% in photon noise or a factor of 1.78 in exposure time. This leads to better constraints on the semi-amplitude and eccentricity of Proxima d, which was recently proposed as a planet candidate. Finally, we applied our telluric model to the O₂ γ-band and we obtained residuals below the 2% P2V amplitude. Conclusions. We propose a simple telluric model for high-resolution spectrographs to correct individual spectra and to achieve precise RVs. The removal of micro-telluric lines, coupled with the gain in spectral range, leads to more precise RVs. Moreover, we showcase that our model can be applied to other molecules, and thus to other wavelength regions observed by other spectrographs, such as NIRPS.