The 68-year-old was due to name the 23 players selected for the tournament this Thursday, but has decided to wait until the conclusion of the Premier League season.
A statement from the Football Association read: "Following a meeting at the FA with his coaching staff this afternoon, the Three Lions boss decided to switch his announcement from this Thursday in order to better assess players' fitness after the final games of the Premier League season."
Arsenal striker Danny Welbeck is expected to miss out on a spot as he is reportedly facing a five-month layoff with a knee injury after getting injured in last weekend's 2-2 draw with Manchester City.
Welbeck's Arsenal teammate Jack Wilshere, though, is vying for a place after he recently returned to action following an eight-month spell on the sidelines with a broken leg.
The Three Lions will play Turkey, Australia and Portugal in international friendlies before their opening Euro 2016 group game against Russia on June 11.