The 22-year-old helped the Greek giants complete the domestic double this season and is now reportedly attracting interest from a number of top European clubs, including the Toffees.
The Hungary Under-21 international is entering the final year of his contract at the Karaiskakis Stadium and boss David Moyes is understood to be looking for new competition for Tim Howard.
Reports indicate Megyeri would cost in the region of £800,000 to be prized away and representative Janos Hrutka said: "Of course, he'll have the last word on his future, but it's important for him to be playing in Europe and to have an opportunity of playing first-team football.
"He's young so the financial side is not the most important aspect of any potential move and even staying at Olympiakos could be an option.
"However, he could easily change clubs if it meant a step forward in his career."
Borussia Dortmund hears Wembley Calling and are headed to London for the Champions League final.
The push for more color on their cleats, adidas has added the infrared colorway to the adiZero f50 Line.
The new PUMA evoSPEED 1.2 FG. A great new soccer cleat with classic look and first time for PUMA to be on the cleat.
Ajax has released their away soccer jersey for the 2013/14 season. Get in the Gear.
Chelsea won the Europa League on Wednesday. The Blues defeated Benfica 2-1 on injury time goal from Ivanovic.
One of the top soccer cleats for control the Nike CTR360 will not let you down on the field in tight spaces.
The Juventus jersey for the 2013/14 was released earlier this week. Be the first to wear the colors of the Old Lady.
Bayern Munich are kings of Germany and hope to be crowned kings of Europe at Wembley later this month!
The Liverpool 2013/14 soccer jersey is available for pre-order. The Reds are eager to compete for silver next season.
adidas has added some color to the Predator line with this new colorway. Released on May 17. Be the first to own a pair.