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The Premier League club said Buvac remained a Liverpool employee but is spending time away from the first team for personal reasons.
Bosnian Buvac played with Klopp at Germany's Mainz in the 1990s and was his assistant manager at the club before moving with him to Borussia Dortmund and on to Liverpool in 2015.
Liverpool's position is that the club consider the matter to be private and they will not be making any further comment.
Liverpool lead Roma 5-2 from the first leg at Anfield and if they succeed in defending that advantage at the Stadio Olimpico they will face Real Madrid or Bayern Munich in the final in Kiev on May 26.