The reigning champions lost 3-1 to Southampton on Saturday evening, meaning league leaders Manchester United could go 12 points clear if they beat Everton on Sunday.
Mancini was as long as 4/1 before the shock loss, but City are without a win in their last three games and face Chelsea in their next league encounter.
Sky Bet's Chris Spicer said: "The pressure is mounting on Roberto Mancini. Ever since City's Champions League exit his every move has been scrutinised, and now it looks as though the league is out of reach too.
"City overcame a similar situation last season, but football can be ruthless, and we're not convinced the club's owners are sentimental enough to stick with Mancini this time."
Meanwhile, Aston Villa boss Paul Lambert has drifted to a massive 12/1 price to be the next to go, while Rafa Benitez (6/1) appears to have bought himself some more time following Chelsea's win over Wigan.