Riyad Mahrezfailed to a convert a late penalty that would have won the game for PepGuardiola"s side, who were 2-0 down at halftime at the London Stadium.
The resultleft City on 90 points with one game to play and second-placed Liverpool,chasing an unprecedented quadruple of trophies, four points behind but with twomatches left.
Liverpooltravel to Southampton on Tuesday and victory in that game would leave them onepoint behind heading into Sunday"s final round of games. A defeat would handthe title to City.
Both sidesare at home for their last games - City face Aston Villa, managed by formerLiverpool favourite Steven Gerrard, while Juergen Klopp"s team hostWolverhampton Wanderers.
Jarrod Bowenstruck twice for West Ham in the first half to stun City but Guardiola"s menfought back strongly.
JackGrealish fired City back into contention four minutes after the break and theHammers" Czech defender Vladimir Coufal headed a Mahrez free kick into his owngoal.
City had agreat chance to take all three points when a VAR review led referee AnthonyTaylor to award a penalty for Craig Dawson"s challenge on Gabriel Jesus.
Mahrez struckthe spot-kick firmly but at a good height for Lukasz Fabianski who produced anexcellent save to deny the Algerian, frustrate City and keep Liverpool"s hopesalive.
TottenhamHotspur boosted their hopes of Champions League football next season with anervy 1-0 win at home to relegation-threatened Burnley.
Spursclimbed above North London rivals Arsenal into fourth on 68 points from 37games -- the Gunners are on 66 from 36 matches and can return to the top fourif they win at Newcastle United on Monday.
CONTROVERSIALPENALTY
The game wasdecided by a controversial penalty just before the break when referee KevinFriend reviewed a handball incident and ruled that the arm of Ashley Barnes hadbeen in an "unnatural position" when the ball struck it.
Kane drilledthe spot-kick into the bottom corner but Burnley"s protests continued after thehalftime whistle with angry exchanges in the tunnel.
Nick Popemade two outstanding saves after the break to keep out Son Heung-min whileBarnes struck the post for Burnley with a thundering drive from distance.
Burnley"sdefeat left them back in the bottom three but the battle to avoid relegationalso looks likely to be decided on the final day of the season.
Leeds Uniteddropped two points with a 1-1 draw at home to Brighton & Hove Albionalthough they were delighted to have salvaged a point with a stoppage-timeheader from substitute Pascal Struijk.
FormerManchester United and Arsenal striker Danny Welbeck had put Brighton ahead inthe 21st minute and Leeds laboured before Struijk snuck in at the back post toset off wild celebrations.
Evertonslipped back into trouble after a 3-2 home defeat to Brentford, ending the gamewith nine men after receiving two red cards.
DominicCalvert-Lewin"s first goal since August put Frank Lampard"s Everton ahead buttheir day turned sour when teenage centre back Jarrad Branthwaite was sent-offfor a foul on the last man.
Brentfordtook control after that as Yoane Wissa forced Everton captain Seamus Coleman toput the ball into his own net, though Richarlison restored the home advantagefrom the penalty spot to give his team a 2-1 lead at halftime.
But twoheaded goals in three minutes just past the hour-mark from Wissa and RicoHenry, and a late red card for Salomon Rondon, handed Brentford a win thatleaves Everton still in need of points for safety.
Everton arein 16th place on 36 points with two games left to play. Leeds are a pointbehind with one game left and Burnley are in 18th, inside the drop zone, on 34points with two matches left.
Burnleytravel to Aston Villa before their final game at home to Newcastle. Leeds"slast match is away to Brentford. Everton host Crystal Palace before travellingto Arsenal next Sunday.
