The midfielder is down to the final year of his contract at the Stadio Olimpico and until recently it had seemed extremely unlikely he would sign an extension.
However, the options are getting whittled down rapidly, as Sky Sport Italia claim Juve and Inter have both given up on Milinkovic-Savic for the moment, unless Lazio lower their asking price.
They consider €40m to be over-priced for a player in the final months of his contract and are refusing to pay up.
Inter already secured Davide Frattesi from Sassuolo, while Juve kept hold of Adrien Rabiot.
The only club willing to pay the sum requested for Milinkovic-Savic are in the Saudi Pro League, but the midfielder is not prepared to make that move.