Il Messaggero has claimed Jurgen Klopp is targeting Milinkovic-Savic as a replacement for Dutchman Gini Wijnaldum.
The latter has sealed a free transfer move to Paris Saint-Germain for the next few years.
The 26-year-old is likely to command a fee in the region of €60 million.
The all-action midfielder had 17 goal involvements for Lazio last season.
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