Enzo Fernandez poked home Chelsea's equaliser in the 94th minute at the Etihad to seal a 1-1 draw on Sunday.
City led through Tijjani Reijnders at halftime, but they are now six points behind leaders Arsenal, who beat Bournemouth 3-2 on Saturday.
Chelsea bounced back from the departure of Club World Cup-winning coach Enzo Maresca to hurt City's title aspirations.
With Strasbourg coach Liam Rosenior reportedly the favourite to take over, youth coach Calum McFarlane stepped up as interim for Sunday's game.
Reijnders showed good feet before unleashing a powerful left-footed shot to beat Chelsea keeper Filip Jorgensen at his near post.
City had chances to extend the lead and paid the price for not taking them when Fernandez arrived at the far post to level the game.
It was a second successive draw for City after being held by Sunderland earlier in the week and they were also hit by injuries to defenders Josko Gvardiol and Ruben Dias.
"It doesn't look good," City manager Pep Guardiola said when asked about the duo.
At Craven Cottage, Harrison Reed scored a spectacular goal in stoppage time to earn Fulham a 2-2 draw with defending champions Liverpool.
The substitute let fly from around 25 metres to beat Liverpool goalkeeper Alisson in the seventh minute of time added on at Craven Cottage.
It stunned Liverpool fans, who were still celebrating after Cody Gakpo had scored what looked like a surefire winner in the 94th.
But they were left stunned by Reed's near instant response once play resumed, with his strike being one of the goals of the season.
Fulham had led 1-0 at halftime through Harry Wilson, who burst through and fired low into the far corner, before Florian Wirtz equalised in the 57th minute.
The draw ensured Liverpool dropped yet more points in what has been a difficult second campaign for coach Arne Slot, as they now lie fourth, 14 points behind Arsenal.
"I am experiencing this for months now that the first chance the other team gets, and the only chance they got in the first half, led to a goal," said Slot. The second thing we see a lot is that the other team scores a goal you don't expect in extra (stoppage) time, so this is then a surprise."
Matheus Cunha rescued a point for Manchester United in a 1-1 draw at Leeds, the Brazil forward equalising after American Brenden Aaronson had put the home side ahead at Elland Road.
Aaronson opened the scoring in the 62nd minute on Sunday, but Cunha levelled the scores within three minutes.
The Brazilian even came close to finding a winner late in the match with a curling shot from outside the box that struck the outside of the post.
Manchester United have lost just two of their last 14 games in the league, but six draws during that run have seen Ruben Amorim's team struggle to keep pace with the top three in the standings.
Leeds are unbeaten in seven games and went ahead when Aaronson raced past United defender Ayden Heaven and shot low into the far corner.
Cunha scored his third goal in his last five games when latching onto a pass from substitute Joshua Zirkzee and sliding a shot past Leeds goalkeeper Lucas Perri.
Igor Thiago hit a hat trick in Brentford's 4-2 win at Everton, the Brazilian forward taking his league tally to 14 this season, second behind only Manchester City's Erling Haaland.
Newcastle beat Crystal Palace 2-0 while Tottenham were denied victory at home to Sunderland when Brian Brobbey scored an 80th-minute equaliser to earn the visitors a 1-1 draw.
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