The Hammers had a bid for the 30-year-old forward rejected by the Saints in the January transfer window.
However, they will now return with an increased offer at the end of the season, according to The Mirror.
Lambert recently claimed that he wanted to finish his career at Southampton.
Restore Data
| Team | P | W | D | L | F | A | GD | PTS | |
| 1 | Liverpool | 23 | 17 | 5 | 1 | 56 | 21 | 35 | 56 |
| 2 | Arsenal | 24 | 14 | 8 | 2 | 49 | 22 | 27 | 50 |
| 3 | Nottingham ForestNott'm Forest | 24 | 14 | 5 | 5 | 40 | 27 | 13 | 47 |
| 4 | Chelsea | 24 | 12 | 7 | 5 | 47 | 31 | 16 | 43 |
| 5 | Manchester CityMan City | 24 | 12 | 5 | 7 | 48 | 35 | 13 | 41 |
| 6 | Newcastle UnitedNewcastle | 24 | 12 | 5 | 7 | 42 | 29 | 13 | 41 |
| 7 | Bournemouth | 24 | 11 | 7 | 6 | 41 | 28 | 13 | 40 |
| 8 | Aston Villa | 24 | 10 | 7 | 7 | 34 | 37 | -3 | 37 |
| 9 | Fulham | 24 | 9 | 9 | 6 | 36 | 32 | 4 | 36 |
| 10 | Brentford | 24 | 10 | 3 | 11 | 43 | 42 | 1 | 33 |
| 11 | Brighton & Hove AlbionBrighton | 24 | 8 | 9 | 7 | 35 | 39 | -4 | 33 |
| 12 | Crystal Palace | 24 | 7 | 9 | 8 | 28 | 30 | -2 | 30 |
| 13 | Manchester UnitedMan Utd | 24 | 8 | 5 | 11 | 28 | 34 | -6 | 29 |
| 14 | Tottenham HotspurSpurs | 24 | 8 | 3 | 13 | 48 | 37 | 11 | 27 |
| 15 | West Ham UnitedWest Ham | 24 | 7 | 6 | 11 | 29 | 46 | -17 | 27 |
| 16 | Everton | 23 | 6 | 8 | 9 | 23 | 28 | -5 | 26 |
| 17 | Wolverhampton WanderersWolves | 24 | 5 | 4 | 15 | 34 | 52 | -18 | 19 |
| 18 | Leicester CityLeicester | 24 | 4 | 5 | 15 | 25 | 53 | -28 | 17 |
| 19 | Ipswich TownIpswich | 24 | 3 | 7 | 14 | 22 | 49 | -27 | 16 |
| 20 | Southampton | 24 | 2 | 3 | 19 | 18 | 54 | -36 | 9 |
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