Leverkusen are thought to be willing to allow the 21-year-old to leave the club at right price in the summer.
However, the Toffees will face competition for his services from Bayern Munich, according to agent Tobias Sander.
"Fabi has got four alternatives," Sander told Bild. "Bayern Munich, Fortuna Dusseldorf, SpVgg Greuther Furth and Everton.
"Bayern Munich definitely has a special status. The training facilities, as much as the atmosphere at the club, are just a class in their own."
Giefer made his Leverkusen debut in 2009.