Pellegrini is to be replaced by Pep Guardiola at the end of the season but could now leave having failed to secure a top-four finish after City's quest to clinch a Champions League berth suffered a setback with a 2-2 draw with Arsenal on Sunday.
Sergio Aguero and Kevin De Bruyne twice gave City the lead, only for the hosts to be pegged back on each occasion by Olivier Giroud and Alexis Sanchez.
The result means Manchester United can now leapfrog City into fourth place if they win their game in hand away at West Ham on Tuesday.
Fans quickly departed the Etihad Stadium following the full-time whistle, with smatterings of supporters remaining to applaud Pellegrini - who has won one league title and two League Cups in his time with the club - and his players.
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Speaking on the pitch afterwards, Pellegrini said: "It was not our way that we wanted to finish the season, we wanted to qualify for the final of the Champions League and win today [Sunday] for next season's Champions [League].
"[I've had] three wonderful years, unforgettable years, all I can say for the fans is thanks for all, I will never forget you and don't change.
"I'm sure next season you will be very successful in all the competitions. Thanks for all and I see you."