The Sun says United had scouts posted last week to check on the youngster in Malmo's Europa League defeat to Panathinaikos.
Malmo lost 1-0, though Ellborg saved a penalty on the night.
It's understood United goalkeeping scout Tony Coton has been tracking Ellborg for some time.
Indeed, United football director Simon Wilcox recently revealed Senne Lammens' signing was done after a year's background work completed by Coton.
Wilcox said: “He was relentless with it. This guy is going to be a top signing for us and you know he started well, but he’s got to keep it going.
“He’s very reflective. He’s very professional, his approach. He wants to get better.”
United see Ellborg as cover for Lammens with Andre Onana, currently on-loan with Trabzonspor, to leave next year, while Altay Bayindir is also expected to depart.