However, the Anfield club maintain Buvac remains a employee of the club.
Buvac is said to have complemented Klopp's more impulsive urges with a more cerebral approach, a tactician and training-ground technician who worked on attacking positional drills with players.
Liverpool have rubbished the rumour by saying that Buvac is "spending some time away from the first-team environment between now and the end of the season" and declined to comment further on what they regard as a private matter.
Along with Peter Krawietz, they have formed a close-knit triumvirate who arrived at Liverpool together in late 2015.
However, the a report in the Times has speculated that the departure may be made permanent a the end of the campaign, in what would be a huge blow to Klopp and the Liverpool backroom team as a whole. He's the one I've learned the most from.
"If it is necessary, I get up from the bench and that is okay", he said. The Reds are also braced for the departure of U18s coach Steven Gerrard, with Rangers hoping to confirm the Kop legend as their new boss this week.
2005 Champions League victor with Liverpool Hamann said Buvac's absence was "huge news".
The Reds hold a 5-2 advantage from last week's first leg at Anfield and must avoid a three-goal defeat at Stadio Olimpico to be certain of reaching the final, which will be against either holders Real Madrid or Bayern Munich. Captain Jordan Henderson and defender Joe Gomez suffered ankle knocks against Stoke, while Trent Alexander-Arnold was substituted due to a dead leg. Sadio Mane missed the game due to a thigh issue, though all four are expected to make the trip to Italy.