City were level with Barclays 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.
But despite being all but out of the title race, Press Association Sport understands 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.
''We are not thinking about Chelsea, or about the title
We are just trying to win our games.''
Source : PA
Source: PA