The Evening Standard claim manager Ole Gunnar Solskjaer has told executive vice-chairman Ed Woodward that he wants all of his summer transfer business concluded by July 1.
James, 21, will reportedly cost between £15-20million and signing him means they will miss out on Borussia Dortmund sensation Jadon Sancho who was their top target.
The Welshman, though, represents a much cheaper option as Sancho, 19, would cost closer to £80m.
It is claimed Solskjaer believes he could solve their problem on the right wing.
James has scored four goals and has seven assists in the Championship this season and possess blistering pace, something that is lacking in United’s current squad.
He has played primarily on the left for the Swans this term but is equally adept on the other flank as he is naturally two-footed.
Solskjaer was asked about the move before United signed off their season with a 2-0 defeat by Cardiff at Old Trafford and refused to rule it out, even if he did not name the player specifically.
“I don’t know how many names have been mentioned,” he said.
“There are players I have barely seen who it says we are going to sign.
“The earlier you can do the transfers, the better it is.
“I can’t comment on every single individual.
“I don’t like when others talk about my players, I won’t talk about others.”
The report from the Evening Standard also claims Solskjaer is keen on Crystal Palace right-back Aaron Wan-Bissaka, 21.
Meanwhile, Express Sport understands West Ham star Declan Rice, 20, is another the Norwegian wants to bring in.
Others who could be on the way out include Paul Pogba who has been heavily linked with Real Madrid.
Romelu Lukaku could also move to Italy and if he goes United will need to make more attacking additions.