The 23-year-old has risen to prominence after taking Serie A by storm after notching 12 goals and providing three assists across the league last season, cementing himself as one of Europe's leading midfielders.
The talent shown by the Serbia international has led to a fierce battle to secure his signature with the likes of United, Real Madrid, Paris Saint-Germain, Chelsea and Manchester City all keeping tabs on his remarkable rise.
And according to Mundo Deportivo, Barcelona have entered the fray and are pushing hard to secure a deal having 'spied' on Savic on numerous occasions throughout last season.
The report claims the Catalan giants could have the edge over their rivals as Savic holds a Spanish passport having been born in the region.
The 23-year-old is contracted to Lazio until 2022 and as a result a mammoth bid would be needed to prise him away from the Italian outfit, with fees from €80m to €150m reportedly mooted in order to sanction a deal.
Savic's unique style of play has attracted the interest of Barcelona as they look to bolster their midfield and attack in the wake of losing Andres Iniesta and missing out on Atletico Madrid forward Antoine Griezmann.
On a side note, how good is Sergej Milinkovic-Savic? He's absolutely phenomenal. He makes the game look unbelievably easy. In all aspects of the game he's incredibly smooth. Wherever he ends up this summer, somebody has got a genuinely world class midfielder on their hands. pic.twitter.com/B5TLb91t6B— Matt (@FalseFirmino) June 17, 2018
However, whilst the media continues to speculate where Savic's next destination could be his international teammate Dusan Tadic has insisted that the midfielder must remain solely focused on their World Cup campaign.
He said, as quoted by the Daily Star: "He is a good player, he is young. He can learn a lot of things but he has a big talent. He has the capacity to go far. That's it. He is the future of Serbian football.
"I say to him to just be calm and work smart and he does. He needs to work for his team and then after he can show his qualities.
“He is a good boy and he just needs to keep his head calm [despite the rumours]. There is a big future for him."
The Catalan giants have also been linked with making a move for Liverpool target Nabil Fekir after his transfer to Anfield failed to be finalised due to concerns over a previous knee injury.