United are looking for a young forward this summer and 22-year-old Nunez is high up on their list of players.
Liverpool have also registered their interest and sources have told ESPN that United would not be put off even if it meant going head-to-head with the Anfield side.
Nunez, according to sources, is keen for regular Champions League football which would put Liverpool ahead in the race but United believe there is a chance he could be convinced to move to Old Trafford.
Sources have told ESPN that an issue for both United and Liverpool is Benfica's valuation.
The Portuguese side initially quoted more than £110 million for the Uruguay international although there is a feeling any bid over £80m would be taken seriously given Nunez, who has a contract until 2025, is keen on a move.
Sources have told ESPN that a number of Spanish clubs are also monitoring the situation.
Liverpool, according to sources, have not yet made a bid for Nunez, who scored 34 goals in 41 games last season, and would be reluctant to get into a bidding war with United, even if Sadio Mane seals a move to Bayern Munich.
Sources have told ESPN that United have prioritised strengthening their forward line and midfield during the summer window, ahead of bringing in reinforcements in defence.
United have what is being described as a "sizeable" budget to work with this summer but the club have distanced themselves from Ralf Rangnick's suggestion while he was interim boss that they could sign 10 players ahead of the new season.
Erik ten Hag is keen to bring in new faces but has also been told he will have to get more out of the players already in the squad.
The Dutchman is on holiday but will return to work before the first squad members return to preseason training on June 27.