Per Football Italia, the midfielder was linked with a €90 million (£83.5 million) move to United this summer, while Paris Saint-Germain are also rumoured to be interested in him.
He told Serbian outlet Espreso (h/t Football Italia):
"The Premier League transfer window is shut, so the Manchester United story is finished. I wasn't irritated by everything that was written in the papers, because my mind was concentrated on preparing for the new season in the best possible way.
"I am where I am and I want for nothing at Lazio. I don't think much about the transfer market. All sorts of things were written about me, but I know where my mind is and who I have a contract with."
At the end of July, Lazio president Claudio Lotito said the club had not received a notable offer for the 24-year-old, per Fabrizio Romano of the Guardian and Sky Italy:
United saw Ander Herrera head to PSG on a free transfer this summer, while Paul Pogba's future has been the subject of much speculation, but the deadline for them to sign new players passed on Thursday without them adding to their midfield options.
Milinkovic-Savic could have added a great deal at Old Trafford.
Last season, he was deployed in a deeper, slightly more defensive role than usual. As a result, his goal and assist totals fell respectively from 14 and seven in the previous campaign to seven goals and three assists.
Nevertheless, as football statistician Dave O'Brien demonstrated, he still returned some impressive numbers:
Statman Dave @StatmanDave
Sergej Milinkovic-Savic vs. other central midfielders in Serie A last season: 1st for aerial duels won (105) 1st for total shots (72) 1st for shots on target (21) 2nd for goals scored (5) 3rd for key passes (45) Big attacking contribution from deep. 🙌 https://t.co/nPupZokeMo
Like Pogba, the Serbian is an impressive blend of imposing physicality and strong technique.
He uses his work rate and strength to help the side out defensively, and he's perhaps even better going forward thanks to his ability to beat an opponent before looking to create chances for his team-mates or convert them himself.
It seems like only a matter of time before he makes the step up to a higher calibre of club than Lazio.
If he remains at the Serie A outfit when the European transfer deadline closes on September 2, it would come as little surprise if the Red Devils were to reignite their interest in him further down the line, particularly if Pogba does move on.