If they lose, then it will again open the door for neighbours Manchester United to go above them and snatch the last top-four place.
Swansea might be a good opponent for City as they have little to play for, but the Welsh side are in good form and chasing a top-half finish.
Francesco Guidolin's men have won their last two league matches, scoring seven goals in the process against top-half sides Liverpool and West Ham United, to move up to 11th.
At home, the Swans have won four successive games for the first time in the Premier League, also conceding just three goals in their last seven home league matches.
Guidolin has got a new deal and midfielder Leroy Fer, currently on loan from Queens Park Rangers, may join him in doing so.
The Dutch midfielder has made 10 league appearances since joining Swansea in the January transfer window, with Guidolin describing him this week as a "good surprise".
Recent form: DWLLWW
He has secured three trophies in three years, including one Premier League title, but Pellegrini's final year has been a tough one.
The announcement that Pep Guardiola was taking his place midway through the campaign did not make things easy and City's league form has suffered, although they did get to the Champions League semi-finals for the first time before losing to Real Madrid.
The Citizens have only won five of their last 13 league matches, but they are still just ahead of rivals United after they lost 3-2 at West Ham United in midweek in the last-ever match at Upton Park.
After last weekend's 2-2 draw with Arsenal, United knew that winning their last two matches would get them into the top four but now the ball is back in City's court.
Their goal difference is far better than the Red Devils', so just a draw is needed this weekend to ensure that Guardiola does not arrive at a club absent from Europe's elite competition. Any slip-up would give United, who play Bournemouth at home, the chance to leapfrog City.
Sergio Aguero has scored in six Premier League games in a row, his longest ever run in the competition, and he is eyeing the golden boot prize as he sits one goal behind Harry Kane (25) ahead of the final day.
While Aguero continues to impress, Wilfried Bony has not proved to be a good acquisition so far for City.
However, the Ivorian has scored in both of his Premier League matches against his former club Swansea. Bony also scored three goals in his last two appearances in this fixture for Swansea against Man City.
City have won only one of their six final-day Premier League games that they have played away from home (D2 L3).
Recent Premier League form: WWDWLD
Recent form (all competitions): DWDLLD
Pellegrini's biggest decision will probably be whether to play Yaya Toure or Kelechi Iheanacho. The former might just get the nod given his experience and the fact that he has scored four goals in his last four Premier League games against Swansea.
Swansea possible starting lineup:
Nordfeldt; Naughton, Fernandez, Amat, Kingsley; Fer, Cork, Britton; Barrow, Ayew, Routledge
Man City possible starting lineup:
Hart; Sagna, Otamendi, Mangala, Clichy; Fernandinho, Fernando; Navas, Toure, De Bruyne; Aguero
Head To Head
Swansea have only won one of the nine Premier League meetings, losing seven.
One of those defeats was a cruel one in the December reverse when they appeared to grab a late equaliser only for City to go up the other end and score.
We say: Swansea 0-2 Man City
There is so much at stake for City here and even though they couldn't get the job done against Arsenal, Swansea should not prove as difficult a challenge.