The 34-year-old has been without a club since leaving Valencia at the end of last season, but the 32-time Argentina international offers a wealth of experience.
Spanish outlet Sport reports that the centre-back has been offered to the two European giants as a solution to their problems at the back.
Liverpool have lost the influential Virgil van Dijk to a long-term injury, while Joe Gomez, Fabinho and Trent Alexander-Arnold have also had fitness issues.
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Barcelona’s defensive situation isn’t much better as Gerard Pique has been ruled out for several months, adding to an injury list that includes Samuel Umtiti, Ronald Araujo, Sergio Busquets and Sergi Roberto.
Veteran defender Garay could be a low-cost solution to the two clubs’ problems as he looks for a new challenge.
Barcelona’s top January target remains Manchester City defender Eric Garcia, although the Premier League side’s asking price is too high for a player who is out of contract next summer.
That could push Barca towards considering a move for Garay, but they might face competition from Anfield.
The Argentine had spells with Real Madrid, Benfica and Zenit before his four-year spell at Valencia, making more than 200 La Liga appearances and regularly featuring in the Champions League and Europa League.
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