The 27-year-old is understood to have expressed his desire to leave the Serie A club this summer, after eight years in the Italian capital.
Milinkovic-Savic, who recorded 11 goals and 12 assists in 47 games for Lazio last season, has caught the eye of a number of top European clubs including Premier League trio Arsenal, Chelsea and Manchester United.
According to Il Messaggero, via The Sun, the Gunners have seen an opening offer of €55m (£47.2m) for the Serbia international rejected by Lazio, with club president Claudio Lotito demanding at least €65m (£55.8m) for the midfielder.
The report adds that Arsenal remain in 'constant contact' with Milinkovic-Savic's agent Mateja Kezman and are planning to table a second offer, but they are currently reluctant to meet Lazio's asking price.
Meanwhile, Lazio head coach Maurizio Sarri has allegedly offered Milinkovic-Savic to former club Chelsea and is keen to recruit both Ruben Loftus-Cheek and Emerson Palmieri in exchange.