Toffees boss David Moyes is thought to be keen on providing current number one Tim Howard with some competition at the club for the next season.
The Merseysiders could now land the Hungary Under-21 international for around £800,000, according to The Mirror.
Megyeri's current deal will expire at the end of next season.
Restore Data
| Team | P | W | D | L | F | A | GD | PTS | |
| 1 | Arsenal | 34 | 24 | 5 | 5 | 82 | 26 | 56 | 77 |
| 2 | Manchester CityMan City | 33 | 23 | 7 | 3 | 80 | 32 | 48 | 76 |
| 3 | Liverpool | 34 | 22 | 8 | 4 | 75 | 34 | 41 | 74 |
| 4 | Aston Villa | 34 | 20 | 6 | 8 | 71 | 50 | 21 | 66 |
| 5 | Tottenham HotspurSpurs | 32 | 18 | 6 | 8 | 65 | 49 | 16 | 60 |
| 6 | Manchester UnitedMan Utd | 33 | 16 | 5 | 12 | 51 | 50 | 1 | 53 |
| 7 | Newcastle UnitedNewcastle | 33 | 15 | 5 | 13 | 69 | 54 | 15 | 50 |
| 8 | West Ham UnitedWest Ham | 34 | 13 | 9 | 12 | 54 | 63 | -9 | 48 |
| 9 | Chelsea | 32 | 13 | 8 | 11 | 61 | 57 | 4 | 47 |
| 10 | Bournemouth | 34 | 12 | 9 | 13 | 49 | 60 | -11 | 45 |
| 11 | Brighton & Hove AlbionBrighton | 33 | 11 | 11 | 11 | 52 | 54 | -2 | 44 |
| 12 | Wolverhampton WanderersWolves | 34 | 12 | 7 | 15 | 46 | 54 | -8 | 43 |
| 13 | Fulham | 34 | 12 | 6 | 16 | 50 | 54 | -4 | 42 |
| 14 | Crystal Palace | 34 | 10 | 9 | 15 | 44 | 56 | -12 | 39 |
| 15 | Brentford | 34 | 9 | 8 | 17 | 52 | 59 | -7 | 35 |
| 16 | Everton | 34 | 11 | 8 | 15 | 36 | 48 | -12 | 33 |
| 17 | Nottingham ForestNott'm Forest | 34 | 7 | 9 | 18 | 42 | 60 | -18 | 26 |
| 18 | Luton TownLuton | 34 | 6 | 7 | 21 | 47 | 75 | -28 | 25 |
| 19 | Burnley | 34 | 5 | 8 | 21 | 37 | 69 | -32 | 23 |
| 20 | Sheffield UnitedSheff Utd | 34 | 3 | 7 | 24 | 33 | 92 | -59 | 16 |
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