The 20-year-old is under contract at Partizan until 2030, with the Serbian club reportedly seeking a fee of around £4.7m for him this summer, as well as a sell-on clause.
Rangers recently submitted a bid of around £3.9m for Dragojevic, which was quickly rebuffed by Partizan.
United, meanwhile, have moved to complete a separate deal for Swansea City goalkeeper Kit Margetson.
Margetson, 19, is the son of former Manchester City goalkeeper Martyn Margetson.
Swansea made an offer to keep the teenager, but he is out of contract and is now poised to join United once a Premier League compensation fee is determined, reports the Daily Mail.
Martyn Margetson left his role as Swansea's head of goalkeeping at the end of last season.
The younger Margetson, a Wales Under-21 international, joined Swansea at Under-15 level.
He has not featured for Swansea's first team but played regularly on loan at Connah's Quay Nomads last season.
Manchester United remain in discussions over another senior goalkeeper, with talks having taken place with Sam Johnstone and Karl Darlow.
They are also waiting on Besiktas, or other potential suitors, to complete a move for Altay Bayindir.
Andre Onana has separately signed his paperwork to complete a season-long loan to Trabzonspor, with no option or obligation for the Turkish club to buy him.