Context. The atmospheric absorption of the Earth is an important limiting factor for ground-based spectroscopic observations and the near-infrared and infrared regions are the most affected. Several software packages that produce a synthetic atmospheric transmission spectrum have been developed to correct for the telluric absorption; these are Molecfit, TelFit, and Transmissions Atmosphériques Personnalisées pour l’AStronomie (TAPAS). Aims. Our goal is to compare the correction achieved using these three telluric correction packages and the division by a telluric standard star. We want to evaluate the best method to correct near-infrared high-resolution spectra as well as the limitations of each software package and methodology. Methods. We applied the telluric correction methods to CRIRES archival data taken in the J and K bands. We explored how the achieved correction level varies depending on the atmospheric T-P profile used in the modelling, the depth of the atmospheric lines, and the molecules creating the absorption. Results. We found that the Molecfit and TelFit corrections lead to smaller residuals for the water lines. The standard star method corrects best the oxygen lines. The Molecfit package and the standard star method corrections result in global offsets always below 0.5% for all lines; the offset is similar with TelFit and TAPAS for the H₂O lines and around 1% for the O₂ lines. All methods and software packages result in a scatter between 3% and 7% inside the telluric lines. The use of a tailored atmospheric profile for the observatory leads to a scatter two times smaller, and the correction level improves with lower values of precipitable water vapour. Conclusions. The synthetic transmission methods lead to an improved correction compared to the standard star method for the water lines in the J band with no loss of telescope time, but the oxygen lines were better corrected by the standard star method.