Arsenal twice came from behind to deny City and need a point from their final game to secure a Champions League place of their own next season.
City ended the day two points ahead of Manchester United after manager Manuel Pellegrini's final match in charge at the Etihad but their neighbours have a game in hand and will deny them a Champions League place if they win their final two matches.
"We should have won that game and I think we deserved it. We played very well," Pellegrini said.
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"Our first 25 minutes were some of the best here this season. But Arsenal's goals were good," the Chilean added.
"The effort, speed and character was good, and we were playing against a very good team."
City started brightly.
Sergio Aguero rifled a shot into the net from outside the box but Olivier Giroud quickly equalised with a powerful header, his first goal for 16 games.
Kevin de Bruyne ran half the length of the pitch, holding off four defenders, to grab City's second before Alexis Sanchez grabbed Arsenal's equaliser.
Spurs need a point from their final game at Newcastle United to finish above Arsenal for the first time since 1995 and secure their best finish in the top flight since 1963.
Tottenham's traditional frailty was on show at White Hart Lane, however, as they again failed to secure victory from a winning position after Son Heung-min put them ahead after 15 minutes.
Two goals from Steven Davis sealed a Southampton comeback that kept them in the hunt for a Europa League place.
At Anfield, a much-changed Liverpool side containing just three of the side who started against Villarreal in the Europa League on Thursday, easily saw off Watford 2-0 with goals from Joe Allen and Roberto Firmino.
Leicester City collected the Premier League trophy on Saturday after beating Everton 3-1 and are 10 points clear at the top of the table.
(Editing by Ed Osmond)
