Ole Gunnar Solskjaer is keen to replace the creativity offered by Paul Pogba and Scott McTominay.
Both players have suffered significant injuries and could be out of action during a lengthy period.
As the fixtures begin to pile up, with European football returning in February, Manchester United could be forced to sign a midfielder on loan.
Llorente joined Atletico from rivals Real Madrid only last summer for approximately £27 million.
Since joining, the 24-year-old has only played 310 minutes of football in La Liga so far this season.
Llorente finds himself behind the likes of Saul Niguez, Koke and Hector Herrera in the Atletico pecking order.
It's understood that United's head of global scouting Marcel Bout has made trips to Spain this season to identify potential targets.
The Reds are in need of more than one addition this month, as alluded to by Solskjaer himself.
"We are looking at a couple of signings," he said.
"Strengthening here and there, but the core group is there and the attitude is there and the physical robustness."
It is thought that United are more confident of securing a loan move this month rather than focusing too much energy on permanent transfers.
"Not many clubs want to sell players that they would otherwise want to keep in January," Solskjaer said in November.
"Maybe one or two could be a loan deal but that is not a big money thing, it is just to help the team."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.