Champions City have lost six of their last eight games in all competitions and have fallen out of the title race as a result.
Pellegrini's men sit 12 points adrift of league leaders Chelsea following a 4-2 defeat to arch-rivals Manchester United last weekend, a result that only served to increase the speculation over the Chilean's future at the club.
Rumours of soon to be ex-Borussia Dortmund coach Jurgen Klopp taking over at the Etihad Stadium have not helped matters, and now Pellegrini must look to bounce back from the loss at Old Trafford with a number of big names on the sidelines.
Captain Vincent Kompany is in danger of missing the rest of the season with a muscle injury, while Gael Clichy (groin), Stevan Jovetic, Wilfried Bony and James Milner have all been ruled out.
Belgian defender Dedryck Boyata could be in contention to feature, though, after being declared fit following a knee injury.
And former Real Madrid and Malaga coach Pellegrini said: "It's more important than ever to have the fans, the players and the staff together as one. We are one club.
"My pressure is always the same, to win every game for this team to play in the way I like the team to play.
"Last year there was more pressure when we were fighting for the title. The pressure is for us to recover our normal performance and that's not the same as fighting for the title."
West Ham, meanwhile, will be without Diafra Sakho due to a thigh injury sustained in the 1-1 draw with Stoke City last weekend, joining James Tomkins (shoulder) and Andy Carroll (knee) on the sidelines.
After witnessing City's dip in form, West Ham boss Sam Allardyce is keen for his players to put a team who may be looking over their shoulder at Liverpool in the race for the top four under further pressure.
"I'm very surprised that Manchester City have lost four away games on the trot," Allardyce said. "I think that's a huge shock across the football world based on the amount of talent they have. Like ourselves, they can't seem to find a way to win recently.
"I watched them against Crystal Palace and I thought they defended brilliantly after they scored but generally Manchester City are exceptionally good at breaking down defences.
"Hopefully we can test their nerve, at home they are usually very good so we know how tough it'll be."
