We present the results of simulations on the detectability of $O_2$ in the atmosphere of Earth twins around nearby low mass stars using high resolution transmission spectroscopy. We explore such detectability with each of the three upcoming Extremely Large Telescopes (ELTs), i.e. GMT, TMT and E-ELT, and high resolution spectrographs, assuming such instruments will be available in all ELTs. With these simulations we extend previous studies by taking into account atmospheric refraction in the transmission spectrum of the exo-Earth and observational white and red noise contributions. Our studies reveal that the number of transits necessary to detect the $O_2$ in the atmosphere of an Earth twin around M-dwarfs is by far higher than the number of transits estimated by Snellen et al. (2013). In addition, our simulations show that, when accounting for typical noise levels associated to observations in the optical and near-infrared, the $O_2$ A-band at 760 nm is more favorable to detect the exoplanetary signal than the $O_2$ band at 1268 nm for all the spectral types, except M9V. We conclude that, unless unpredicted instrumental limitations arise, the implementation of pre-slit optics such as image slicers appear to be key to significantly improve the yield of this particular science case. However, even in the most optimistic cases, we conclude that the detection of $O_2$ in the atmosphere of an Earth twin will be only feasible with the ELTs if the planet is orbiting a bright close-by (d $le$ 8 pc) M-dwarf with a spectral type later than M3.
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