The 26-year-old has impressed with his performances domestically and on the international arena, and is valued at £30 million.
Marco Silva, the new Everton boss, is yet to make a signing at the Merseyside club, with the Toffees taking a more cautious approach while recruiting players.
The report claims that Silva wants a classy winger in his squad, and Swedish star Forsberg – who played in the World Cup quarter final against England in Samara – is high on the list.