Sunderland reportedly scouted the Denmark international during the recent international break, while West Ham and Crystal Palace are also said to be keen.
Braithwaite, who is of Guyanese descent, joined Toulouse from Esbjerg in 2013 and is enjoying his most productive campaign yet - hitting double figures for the first time since moving to France.
The 24-year-old has spent much of his career to date occupying a central striker's tole, but can also play out wide if required, as well as in the number 10 role.
Swansea added Italian striker Alberto Paloschi to their ranks in January, though the Jacks could make room for Braithwaite by offloading Bafetimbi Gomis this summer, after the Mirror reported in December that the Frenchman was keen to leave the Liberty Stadium.
The Welsh side, who are expected to appoint a full-time manager to replace Francesco Guidolin this summer, have also been linked with a £6 million move for Ecuador forward Miler Bolanos, by the Sun.