The 30-year-old is expected to fall down the pecking order at the La Liga side when their deal for Chelsea forward Diego Costa goes through in January.
According to The Mirror, Atletico are worried about balancing their books after committing to spend £67m on Costa and have "sounded out" Ronald Koeman's side over the possibility of them signing Gameiro.
The Toffees remain in the market for a striker, having sold Romelu Lukaku to Manchester United over the summer, and have been linked with a string of names in recent weeks - including Olivier Giroud, Edinson Cavani, Danny Welbeck and Seydou Doumbia.
Gameiro joined Atletico from Sevilla in July 2016 and enjoyed a successful debut season, scoring 12 goals and creating five more in 31 La Liga appearances, but has been used just twice so far this term.
The Frenchman is expected to be available for around £20m.