The 18-year-old has impressed since joining the Bundesliga high-flyers from Austrian outfit Red Bull Salzburg in January 2017, and is believed to be on the radar of several top European clubs.
The Reds, meanwhile, have already conceded nine goals in the Premier League this term and have been criticised for failing to bring in defensive reinforcements in the summer.
According to German newspaper Bild, Jurgen Klopp has cast his eye back on his homeland and identified the French teenager as a January target to strengthen his defence.
The report, however, goes on to suggest that Upamecano has a new deal on the table at the Red Bull Arena which would raise his buyout clause to the €100m (£88m) mark.
Liverpool have already had dealings with Leipzig this year, having secured the services of midfielder Naby Keita as of next summer.