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Most metal-rich AGB/RGB stars present strong Li underabundances, since this element is easily destroyed in the high temperatures of the stellar interiors. In spite of this fact, several of these stars are Li-rich, having Li abundances given by log (Li/H) + 12 > 1.5. In a previous work we have shown that high-metallicity Li-rich stars follow the same average Li abundance trend with metallicity as the metal-poor stars, although with a larger dispersion. More recently, we have investigated the existence of correlations of the Li abundances with several physical properties of the stars, such as the effective temperature, mass, radius, and luminosity. In the present work, we extend this investigation to the expected mass loss rates of these stars. Specifically, we look for correlations between the Li abundances and the mass loss rates or related parameters in Li-rich AGB/RGB stars. We have estimated the mass loss rates using a modified form of the Reimers formula and applied it to a large sample of 104 Li-rich giant stars for which reliable stellar data are available. Our proposed method assumes a linear relation between the stellar luminosity and the Li abundance, so that the luminosity can be estimated from the Li abundance. The stellar mass is then obtained from the effective temperature and luminosity using recent evolutionary tracks. The stellar radius can be determined from the stellar gravity, so that the mass loss rate can be calculated using an adequate calibration involving both Li-rich and Li-poor stars in the AGB/RGB branches. The results show that most Li-rich stars have lower mass loss rates compared with C-rich or O-rich giants that do not present Li enhancements.