Grujic has struggled to hold down a place in the Liverpool team since arriving from Red Star Belgrade in 2016, making just 14 appearances for the first team over the past two years.
The 22-year-old spent the second half of last season on loan at Cardiff City and helped the Bluebirds to automatic promotion from the Championship before returning to his parent club.
Grujic remains highly rated at Anfield, but The Mirror reports that Lazio could look to test their resolve with a £5m bid.
The player's first-team chances are likely to be further limited next season following the arrivals of Naby Keita and Fabinho, while Liverpool are also understood to be trying to bring in another midfielder.
The Serbia international, whose contract with Liverpool runs until 2020, is part of his country's squad for this summer's World Cup in Russia.