The 28-year-old has stayed loyal to the Serie A giants for the past few years, despite growing interest from a host of Premier League and foreign clubs.
However, Milinkovic-Savic has now entered the final year of his contract with Lazio, who could therefore be pressured into a sale this summer before he departs on a free transfer.
Lazio sporting director Igli Tare recently admitted that the Serbia international could leave in the coming weeks, and the Italian giants are not in a strong negotiating position given his contract situation.
Milinkovic-Savic is represented by former Chelsea attacker Mateja Kezman, who has offered his client to the Blues for a modest fee, according to football.london.
The report states that Chelsea have been presented with the chance to bring Milinkovic-Savic - who has also been linked with Manchester United - to Stamford Bridge for just £34m.
The Blues have lost a host of midfielders since the end of their miserable 2022-23 season, with all of N'Golo Kante, Ruben Loftus-Cheek, Mateo Kovacic and Tiemoue Bakayoko exiting for pastures new.
Mason Mount's move to Manchester United is also expected to be confirmed within the next 24 hours, as the England international is reported to have completed his medical with the Red Devils.
However, the Blues are yet to sign a single midfielder in the current window, only bringing in attacking duo Christopher Nkunku and Nicolas Jackson from RB Leipzig and Villarreal respectively.
Nevertheless, new head coach Mauricio Pochettino has promised that his side will welcome more new faces before the start of the campaign, and Milinkovic-Savic would replace Mount's goal threat from the middle of the park.
The former Genk man has consistently delivered in the final third since joining Lazio in 2015, registering 69 goals and 59 assists in 341 games in all competitions.
Another eye-catching 2022-23 season saw Milinkovic-Savic register 11 goals and eight assists from 47 appearances, including nine strikes from 36 Serie A appearances.
The Serbian has collected one Coppa Italia and two Supercoppa Italiana crowns with the Biancocelesti, and he would also represent a cheaper alternative to Chelsea's other midfield targets.
Brighton & Hove Albion's Moises Caicedo is believed to be at the top of Chelsea's shortlist, but the Seagulls have reportedly slapped an £80m price tag on the head of the Ecuador international.
Southampton's Romeo Lavia has also been linked with a move to Stamford Bridge, but the Belgian could also command a fee of at least £50m.