Gylfi Sigurdsson will see a specialist on Monday and undergo additional tests after he injured his right knee in Everton's 2-0 win against Brighton on Saturday.
The club said in a statement that Sigurdsson "could be sidelined for several weeks but it will only be after consulting with the specialist that a definitive timeframe will be known."
Any significant injury could put the Iceland international's participation in this summer's World Cup, the country's first, in doubt.
Sigurdsson injured his knee in the first half of the victory at Goodison Park but played the full 90 minutes, with manager Sam Allardyce only describing the midfielder's injury as a "knock."
His brother Olafur Mar Sigurdsson spoke to Icelandic media and said the joint of the right knee is damaged.
The £45 million club-record signing endured a difficult start to his time with Everton but his fortunes have improved since Allardyce replaced Ronald Koeman in late November as he has scored five of his seven goals in his 17 matches since.