Milinkovic-Savic, 27, has been an influential asset for the Serie A outfit over the last seven seasons, scoring 58 goals and providing 51 assists in 294 appearances across all competitions.
However, the Serbia international's future in the Italian capital has been questioned for some time, amid interest from top Premier League clubs including Chelsea, Arsenal and Manchester United.
According to Italian news outlet Il Messaggero, via the Daily Express, Lazio are currently in 'turmoil' as Milinkovic-Savic has expressed his desire to leave the club, with the player's agent and former Chelsea striker Mateja Kezman reiterating that his client has no intention of staying at the Stadio Olimpico.
The report adds that former Chelsea boss Sarri has attempted to offer the midfielder to the West London club, but he is keen to recruit midfielder Loftus-Cheek and left-back Emerson in exchange.
Loftus-Cheek may consider a summer exit from the Blues if he is not guaranteed regular first-team football, after starting only 18 of his 40 appearances under Thomas Tuchel last season, while Emerson's future at Stamford Bridge remains uncertain following a loan spell with Lyon in 2021-22.
The report also claims that both Arsenal and Newcastle United have tabled bids between €50m (£43m) and €55m £47.3m) for Milinkovic-Savic, but neither of these offers match Lazio's valuation of the midfielder.