West Ham can finish anywhere between fourth and tenth in their final two games.
Leicester City are now out of reach on 66 points as the Hammers are seven points behind them.
However, there are another six clubs in play to complete for final positions.
West Ham are on 59 points so with two to play their maximum will be 65 points with West Brom and Southampton left to play.
It means the Hammers will finish on 59,60,61,62,63 or 65 points.
Chelsea have two games and are on 64 points, they will finish on 64-70 points with Leicester City and Aston Villa left to play.
West Ham needs Chelsea to lose both games to steal their Champions League fourth place.
Liverpool are now on 63 points after beating West Brom, they will finish on 63-69 points with Burnley and Palace left to play.
Spurs are on the same points as West Ham with a superior goal difference, they will finish on 59-65 points with Villa and Leciester City left to play.
Everton on 56 points with 3 games left to play with Sheffield United, Wolves and Man City leave to play. Their final points will be between 56-65 with an inferior goal on West Ham as it stands.
Leeds United on 53 points with two to play, they have Southampton and West Brom and can finish on 59 points and overcome West Ham only if the Hammers lose both games and they make up a six-goal difference in the progress.