Roberto Mancini's team are on the verge of making an early exit from Europe's elite club competition for the second successive season after taking just one point from their first three Group D matches.
The Premier League champions were already in trouble after losing to Real Madrid and drawing at home with Borussia Dortmund, but it was last month's 3-1 defeat against Ajax in Amsterdam that pushed them to the brink.
Anything but a revenge win over the Dutch champions at Eastlands will effectively end City's challenge for a place in the last 16 and heap more pressure on Mancini, who is becoming an increasingly vulnerable figure as his team splutter both in Europe and in their defence of the English title.
Reports that former Barcelona coach Pep Guardiola is being lined up to replace Mancini next season may be premature, but the Italian knows his hopes of remaining in charge will be severely damaged by another Champions League flop.
To make matters worse for Mancini, who is likely to be without injured England internationals Joleon Lescott and James Milner, City are hardly in peak form heading into their showdown with Ajax.
They failed to score for the first time in seven months on Saturday as West Ham easily held them to a 0-0 draw at Upton Park and now have only four goals in their last four matches.
Fortunately for Mancini, Belgian defender Kompany is adamant City's spirit remains strong and he believes last season's come from behind triumph in the Premier League is evidence they can rescue their European campaign from even this perilous position.
"Last time, we were eight points down with six games to go - the change in attitude was just to say, 'OK we are not doing it for the points anymore, we are not doing it to win the league. We are doing it for our own pride because we know we are better than what we have shown," Kompany said.
"I think the same is going to be needed now. It is not about winning three games to get through, it is us showing to ourselves we belong there and then we will see what happens.
"Man City will be in the Champions League many more times and will get its chance again but now we have to show that we deserve to be in the Champions League."
Last season City were at least partially able to put their elimination from the Champions League down to a lack of experience on that grand stage and Kompany feels the club are still learning how to compete at the elite level.
"It is a challenge, it was never going to be easy. Any quotes from me will just be positive about what this team can achieve," he said.
"I don't want to participate in the whole dramatising of what is happening here.
"Man City will get there, with or without us. This is just a moment in time where we just have to pull together and get ourselves back on track."
Unlike diplomatic skipper Kompany, City's outspoken midfielder Samir Nasri took a more aggressive stance when asked about his side's problems of late.
Nasri admitted City looked out of sorts for long periods against West Ham and he knows it will take a major improvement to avoid another disappointment on Tuesday.
"We should have done something better," he said. "We missed so many chances and when you miss these kinds of chances, you cannot expect to win the game. So we have to improve."
Ajax warmed up for the tie in the same lacklustre manner as City, leaving their title challenge in tatters and Europe as their primary focus.
Frank de Boer's side, who have three points in the Champions League so far, were beaten 2-0 at home by Vitesse Arnhem on Saturday and lie 10 points behind leaders FC Twente.