The 21-year-old prospect is highly sought after, with Barcelona, Real Madrid, Bayern Munich and Inter Milan also monitoring his progress.
As reported by Sport, United are determined to sign Alvarez, who has scored 18 goals in 21 league games this season.
United interim manager Ralf Rangnick is determined to bolster his team’s attacking options, and Alvarez is seen as a player who could help the club in that department.
Alvarez, who progressed through the ranks at River Plate’s academy, is a senior Argentina international and has been capped by his country on five occasions.
Rangnick can already call upon the likes of Cristiano Ronaldo, Edinson Cavani, Mason Greenwood and Marcus Rashford, but United are still looking to strengthen.
Alvarez’s agent Fernando Hidalgo is reportedly set to fly out to England in order to discuss a deal.
So far this season United have scored 30 goals in the Premier League – 21 fewer than leaders Manchester City, who have played two more games.
It has been an inconsistent season for United so far – they are currently in sixth place in the league.
On Monday United host Wolves, with Rangnick’s side looking to build upon their 3-1 win against Burnley.