The 17-year-old Dundee striker has reportedly been the subject of a £200,000 offer.
The Premier League trio are said to have scouted the SPL star extensively in recent months, while Championship side Blackburn Rovers have also reportedly shown interest.
Wighton is rated as one of Scotland's brightest stars and is Dundee's youngest ever goalscorer.
It is Southampton that have stolen a march on Sunderland and Swansea, the Mail claim, though, with a £200,000 offer in the works.
Southampton are famous for their prestigious youth academy and the 17-year-old would no doubt benefit greatly from the coaching setup at St Mary's.
This may not be the last we hear of the Saints' pursuit of Wighton, however, due to Sunderland and Swansea's monitoring of the forward. Dundee, according to the Mail, will find it hard to resist offers.
Sunderland and Swansea have Premier League fixtures coming up at home this Saturday, while Southampton travel to Manchester United on Sunday.