City were level with Premier League leaders Chelsea on New Year’s Day, but they now find themselves fourth in the standings with just seven games to go.
City have lost five of their last seven matches in all competitions and are nine points behind Jose Mourinho’s side, who have a match in hand to come against bottom side Leicester.
However, despite being all but out of the title race, it is understood there is no current appetite to remove Pellegrini, who is contracted until the end of next season.
The Chilean’s performance will be reviewed at the end of the season in line with club policy for all employees.
Pellegrini won the league in his first year at the Etihad Stadium, but his chances of retaining the trophy took a significant blow on Monday when City lost 2-1 at Crystal Palace.
After the defeat, the 61-year-old appeared determined to see out the three-year contract he signed in the summer of 2013.
“I’m not concerned about my job. That is one thing I am never concerned with,” Pellegrini said.
“I never have fear about it, I do my work, I’m very happy. The team may be having a difficult season, but never worry about me.”
Gael Clichy accepts he and his team-mates should have done better at Selhurst Park, even though Palace’s first goal — scored by Glenn Murray — should have been ruled out for an offside against Scott Dann.
“It [the criticism] was deserved,” the City left-back conceded.
“We knew it was going to be a difficult game, because Palace have been playing really well since they had a change of manager, but we also know a team like us, a club like us, with the objectives we have, have to win games like this so we are not hiding behind anything.
“We have heard that some decisions went against us, but we don’t want to hide behind these.
“The results are not there and we need to be better.”
The Frenchman hopes to put a smile back on City supporters’ faces this weekend by claiming victory at the home of their nearest and biggest rivals Manchester United.
“The timing is perfect [for a derby],” Clichy said.
“On Sunday, we are playing against United, they are hitting top form this year and we lost that second position, so we have a lot to make up for to the fans and Sunday night the city will be all blue or red and, hopefully, we can make a great day for them, because the fans have always been behind us.”
Meanwhile, United striker Robin van Persie has declared himself fit for the derby.
Van Persie has missed Manchester United’s last six games following an ankle injury he suffered in the 2-1 defeat at Swansea on February 21.
Manager Louis van Gaal said last week he did not expect the striker to be fit enough to return for Sunday’s game against Manchester City at Old Trafford, but Van Persie reported good news for United fans yesterday.
He tweeted: “Back to training with the group today. Was a great session. Happy to be fit and able to play again. Looking forward to my next game!”