The underperforming English champions are last in Group D with only two points from four matches after opening the campaign with a 3-2 loss at Madrid.
"It is very difficult, because probably we did some mistakes in the first game," City manager Roberto Mancini said Tuesday.
Mancini will make a late decision on whether striker Mario Balotelli's back has recovered in time for him to face the nine-time European champions.
"Yesterday he trained with the physio and tomorrow and today he will train with the team and we will see if he can play," Mancini said.
The pressure on Mancini has been lifted slightly by City rising to the top of the Premier League on Saturday for the first time in its title defence. But it's in Europe where the top teams are judged, and for some City players the last two Group D matches could now only be a matter of pride.
"Unlikely as it is that we will go through, we want to win our remaining two games and give ourselves at least half a chance," City winger James Milner said. "And if that's not good enough, at least we can finish with our heads held high."
Madrid coach Jose Mourinho is already writing off City's hopes of progressing.
"I don't think it is normal when you are out of the competition at the very early stage ... last season they are out, and I think this season they will be out again," Mourinho said. "I think they can win tomorrow. Of course they can. And they can also win in Dortmund. Why not? But eight points is not enough.
"So I think they are out for the second consecutive time and that must be difficult for them."
Madrid is second in Group D, one point behind Borussia Dortmund and three ahead of Ajax.
But Madrid has stumbled in its last two matches, losing at Dortmund and then being held by the German champions at home two week ago.
"We know every mistake costs and we have known since the beginning everything in this group is in risk," Mourinho said. "Other teams are playing easy groups and have known from day one they would qualify easily.
"City, Real Madrid and Borussia all want to win the competition. We all can but we all know we can be out in the group stage, especially as Ajax are also taking points and have some ambitions."
Madrid's passage to the knockout phase would be guaranteed with a game spare with a win in Manchester, where Cristiano Ronaldo will be playing for the first time since leaving United in 2009.
"He does not seem to feel pressure, and there will be no added pressure just because he is back in Manchester," said City striker Carlos Tevez, who played alongside Ronaldo at United. "Once he gets a chance to face up to you and run at you with the ball, he is very, very dangerous. And he strikes the ball well, too."
Ronaldo scored the late winner for Madrid in September against City.
"Tomorrow he should have more attention on him, not only the other players ... because he's a player that can change the game alone," Mancini said.
Winger Denis Cheryshev has been called into the Madrid squad, with midfielder Michael Essien and forward Alvaro Morata sidelined.
Dortmund is the only unbeaten team in Group D and to advance only needs a draw at Ajax, which has never been beaten at home by German opposition.
"I have a lot of respect for Ajax," Dortmund defender Neven Subotic said. "They don't allow themselves to be shaken up."
Ajax will be missing injured striker Kolbeinn Sigthorsson and forward Ryan Babel, who is recovering from a dislocated shoulder.
But coach Frank de Boer said his young Ajax team has gained valuable experience since conceding a late goal in its 1-0 loss in Dortmund.
"It's important that the squad realizes it can get a good result at the very highest level, that we can make life difficult for every team," he said. "You can lose, but you have to go down fighting. We've shown what we can do in three of the four (Champions League) matches.
"Tomorrow I want to see the team oozing self-belief, then it can be an interesting confrontation."