The 27-year-old’s future with the FA Cup and Community Shield champions remains up in the air despite their trophy-laden 18-months under Oliver Glasner.
Eze has been linked with move to Tottenham and Arsenal and is understood to be in talks over a move to Thomas Frank’s Spurs.
Arsenal remain interested, however, but former winger Walcott has warned Eze that a move to the Emirates during a World Cup year may not be wise.
He told It's Called Soccer: "If he has a good season at Palace, he will go to the World Cup, I’m pretty sure of that… then what other teams will have a look at him after that?
“If I was Eze, I would probably stay put just because it’s World Cup year. If it wasn’t World Cup year then I would probably move on. These opportunities might not come again, though, so that’s the risk these players have got to take.
“But for me, World Cup year is a big year and I just feel that the grass isn’t always greener. I love the way Eze plays at Palace at this moment in time and it would look very different at Arsenal.
“He would improve Arsenal, 100 per cent, but I just feel that World Cup year is a big factor for a lot of players this year."