Reports on Friday suggest United are close to agreeing a £15 million ($19.5m) deal to bring the Wales international to Old Trafford ahead of next season.
But speaking at a press conference ahead of his side's meeting with Cardiff City on Sunday, Solskjaer would not be drawn on whether his first signing since taking the reigns at the 20-time English champions would in fact be James.
"I can't comment on every single individual, and there's many, many players," he said when the topic of James was put to him.
"I've said it before: I don't like when other managers talk about my players and I wouldn't talk about any others, even if I wasn't or I was interested.
"I don’t know how many names have been mentioned. There are players I have barely seen who you say we are going to sign! The earlier you can do the transfers, the better it is."
James has impressed during his first full season of senior football at the Liberty Stadium, with his lightning-fast speed causing problems for defences up and down the Championship.
He has scored five goals and laid on 10 assists in 38 appearances in all competitions for the Swans as he helped them to a 10th-placed finish in their first season since relegation from the Premier League.
The 21-year-old's form saw him earn a first cap for Wales in November 2018 before marking his second appearance for his country with the winning goal against Slovakia in a Euro 2020 qualifier in March.
James came close to leaving Swansea during the January transfer window after a £10m ($13m) deal was provisionally agreed with fellow Championship side Leeds United.
The wideman travelled to West Yorkshire and even conducted interviews with the Elland Road outfit's own media team, only for the move to fall through in the late hours of transfer deadline day as the two clubs failed to agree on the structure of the payments.
James is one of a number of wingers to have been linked with a move to Old Trafford this coming summer, most notably Borussia Dortmund's teenage sensation Jadon Sancho.
But reports suggest the 19-year-old is likely to cost in excess of £100m ($130m) while Dortmund have routinely poured cold water on the prospect of Sancho leaving the Westfalenstadion during the summer window.