The 22-year-old is currently on loan at Spanish outfit Getafe and has enjoyed a productive start to life in La Liga following his temporary exit from Old Trafford.
Greenwood has netted six goals and provided five assists in 24 league appearances, while also finding the net twice for Getafe in this season's Copa del Rey.
The forward has made 129 appearances for the Red Devils since making his debut in the summer of 2018 but is now experiencing his maiden stint away from the red side of Manchester.
According to Caught Offside, Barcelona have pulled out of the race to secure the services of Greenwood ahead of the summer trading point.
The report claims that the Catalan club were interested in a move for the former England international, who has caught the eye during his maiden campaign in Spain.
It is understood that Man United directors and Barca chief Deco met for discussions over a possible deal earlier in the season, with the attacker settling in quickly at Estadio Coliseum.
However, it is said that the Camp Nou side have halted their pursuit of Greenwood, with Xavi's side looking towards other targets ahead of the 2024-25 campaign.
Barca are not the only Spanish side who have been linked with the Englishman, though, with Atletico Madrid thought to be monitoring the player's situation at Old Trafford.
Diego Simeone's men are no strangers to having English talent within their ranks, after now-Newcastle United man Kieran Trippier lifted a La Liga crown at Estadio Civitas Metropolitano in 2021.
In a move that sparked controversy, Man United's minority owner Sir Jim Ratcliffe did not rule out
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