The 27-year-old has been persistently linked with a move to the Premier League over the past couple of years, but he has remained loyal to the Biancocelesti since 2015.
Milinkovic-Savic still has two years left to run on his existing deal with Maurizio Sarri's side, but his future is once again being called into question amid links with numerous clubs.
Manchester United were previously tipped to bring the Serbia international to Old Trafford, but the Red Devils are instead focusing their efforts on signing Barcelona's Frenkie de Jong.
Meanwhile, interest from Chelsea is unlikely to go anywhere over the next couple of weeks while Thomas Tuchel prioritises new faces in the back three and attacking line.
Arsenal are therefore understood to have reignited their pursuit of Milinkovic-Savic, but according to Calciomercato, a bid of €50m (£42.3m) was swiftly turned down by Lazio.
The report states that the Serie A side are after at least €70m (£59.2m) to entertain discussions with interested parties, none of whom have indicated that they are willing to match that valuation.
Newcastle United are also thought to have taken an interest in signing the former Genk man, but suitors could instead wait until 2023 to try to strike a deal at a lower price if no new contract is agreed with Lazio.
The Biancocelesti spent just £10.8m to prise Milinkovic-Savic away from Genk in 2015, and the midfielder has since registered 58 goals and 51 assists in 294 games for the club in all tournaments.
The 2021-22 campaign was another productive one for Milinkovic-Savic at club level, as he registered double figures in league goals and assists for the first time in Serie A - scoring 11 and setting up 11 more in 37 outings.
The Spanish-born lynchpin has won one Coppa Italia and two Supercoppa Italianas during his time at Lazio, who have been linked with a move for wantaway Gunners midfielder Lucas Torreira.
The 26-year-old reignited his career on loan at Fiorentina last season, but La Viola ultimately chose not to trigger his £12.5m buy option.
Torreira is now back at London Colney for pre-season training with Arsenal, but he has admitted that he wants to leave after being told that he was not in the club's plans for the new season.
Lazio have supposedly been offered the chance to sign Torreira in the coming weeks, but there is nothing to suggest that the Uruguay international could be used as part of a deal for Milinkovic-Savic.