Swansea are asking for more than £20m for the defender, who is still returning to fitness after undergoing knee surgery last season.
West Ham have had a few offers rejected earlier in the summer, when they bid for both Mawson, and Swans keeper Fabianski.
'Sky Sports' reported that Burnley had opened discussions for the defender, as they look to strengthen their squad ahead of their first season in European football.
Fulham, who have recently signed Maxime Le Marchard, now want another centre-back to add to their ranks.
The Premier League newbies have got involved in discussions last minute.
It's unclear if Swansea or Mawson have a preference for his destination, or whether he will leave the club at all.