Résumé : The existence of dark matter is now well accepted in view of the wide variety of observations that have led to its postulate. Presently, the main objective of dark matter experiments is to identify the nature of this non-visible matter. Assuming that dark matter is composed of massive particles that interact weakly with matter, it is predicted that dark matter will produce Standard Model particles when annihilating or decaying. These Standard Model particles could, in turn, produce stable charged particles found in cosmic radiation, as well as gamma-rays and neutrinos. The Milky Way is expected to be immersed in a dark matter halo with an enhanced density towards its centre. This over density would amplify the probability of dark matter particles to annihilate, making the Galactic Centre an ideal target for indirect dark matter searches. In this thesis, two indirect searches for dark matter annihilation in the Galactic Centre using data collected by two neutrino telescopes are presented.The first analysis is a combined dark matter search using the ANTARES and the IceCube neutrino detectors. By combining a total of ~ 5.8 years and ~ 2.8 years of data collected respectively by ANTARES and IceCube, no neutrino excess was found in the direction of the Galactic Centre and limits on the dark matter annihilation cross-section were set. The limits thus obtained show a considerable improvement compared to the previous results derived separately by the two telescopes, for dark matter masses ranging from 50 GeV to 1 TeV. In order to carry out this first joint analysis, the analysis method as well as the parameters of the different models have been unified, providing a benchmark for future similar searches.In the second analysis, a total of ~ 8.03 years of DeepCore data are used to search for neutrinos coming from dark matter annihilation in the centre of the galaxy at lower dark matter masses. This analysis aims to improve the detection potential for such a search. This low-energy dark matter search allows us to cover dark matter masses ranging from 5 GeV to 8 TeV. The sensitivities obtained for this analysis show considerable improvements over previous results from IceCube and other neutrino telescopes for the entire energy range considered.