Chelsea penned Abraham down to a new five-year deal earlier this month, before he joined Swansea on loan for the forthcoming campaign.
The 19-year-old is seen as one of the Blues’ hottest talents for the future, after shining on loan at Bristol City last season.
Abraham bagged an impressive 26 goals from 48 games for the Championship side, scooping the club’s Player of the Year, Young Player of the Year and Top Goalscorer awards.
However, Antonio Conte has resisted the urge to throw him into his starting line-up this campaign as he wants him to gain Premier League experience.
And Manchester United are now supposedly monitoring his situation with great interest.
All Nigerian Soccer claim the England Under-21 star was watched by a scout dispatched by the Red Devils, Stephen McCarthy, during a game against Birmingham City.
And he did not disappoint, scoring a second goal from three pre-season matches with a fine volley in the 56th minute against Harry Redknapp’s side.
Everton and Aston Villa also sent scouts to the fixture at St Andrew’s, as well as the likes of Queens Park Rangers, Reading and Leicester.
Speaking after his Birmingham heroics, Abraham outlined his desire to score goals galore at the Liberty Stadium.
“I am very pleased,” Abraham said.
“I came here to score goals and help the team as much as I can.
“I have been in the right place at the right time and I am happy I have got my goals,” he added. “It was a nice finish.
“I could have had more but I take that and move on. We get ready for the next game and for the season.”