Stellar activity variability is one of the main obstacles to the detection of Earth-like planets using the radial velocity (RV) method. The aim of this work is to measure the effect of activity in the spectra of M dwarfs and detect activity-sensitive lines in the near-infrared (NIR) to help improve exoplanet detection and characterisation and contribute to further stellar activity analysis in the NIR. We took advantage of the simultaneous observations of HARPS and the newly commissioned NIRPS spectrograph to carry out a blind search of the most activity-sensitive spectral lines in the NIR using NIRPS spectra and known activity indicators in the optical from HARPS as a reference. We analysed the spectra of Proxima (M5.5V) and Gl 581 (M3V), two M dwarfs with different activity levels and internal structures. Spectral lines were identified for both stars and their profiles were fitted using different models. We found hundreds of lines sensitive to activity for both stars; the Proxima spectra were more affected. For Proxima, around 32% of the identified lines can be used to measure the rotation period of the star, while for Gl 581 the numbers drops to 1%. The fraction of lines sensitive to activity increases with increasing line depth for both stars. A list of 17 lines with rotation period detection for both stars is provided. Stellar activity is able to affect a significant number of spectral lines in the NIR, and methods should be developed to mitigate those effects at the spectral level. The line distortions detected here are expected to come mainly from the flux effect due to temperature contrasts between active regions and the quiet photosphere; however, we cannot rule out the possibility that core-emission from chromospheric activity or Zeeman splitting are also affecting some lines. The new line lists presented here can be used to improve the RV extraction and the detection of RV variability due to stellar activity signals, and to help false positive detection and the modelling of activity variability, thereby enhancing exoplanet detection in the NIR.