According to the Daily Mail, Fulham hope to complete a deal for the highly rated centre back early this week.
Mawson was one of few Swansea players to enhance his reputation last season, as he was a consistently impressive performer despite his side's dismal form. In fact, his displays attracted the interest of England manager Gareth Southgate, who included the 24-year-old in the squads to face the Netherlands and Italy in March - although the defender didn't play in either game.
Unfortunately, Mawson was unable to save the Swans from relegation, as they finished in 18th place in the Premier League with a mere 33 points. Now he is hopeful that he can move to London and continue his career in the top flight.
Mawson won't be cheap - he is valued at £25m - but Fulham manager Slavisa Jokanovic is evidently undeterred by this valuation. Compared with the £80m which Leicester City would reportedly require for Harry Maguire, Mawson's price tag makes him look like a potential bargain.
If the Swansea defender joins the Cottagers, they will have achieved two major transfer coups this summer, as they are on the verge of completing a £22m deal for Newcastle United's Aleksandar Mitrovic. The Serbia striker was already on loan at Craven Cottage, where he netted 12 times in 20 matches last season, thus playing a crucial role in Fulham's promotion.
Jokanovic has worked tirelessly this summer to secure Mitrovic's services on a permanent basis, and his efforts seem to have paid off. The Serb left Newcastle's pre-season training camp in Portugal on Saturday and is scheduled to have a medical at Craven Cottage on Monday.
With Mawson at the back and Mitrovic up front, the Cottagers could be a force to be reckoned with in the 2018/19 season.