The Tigers head to the Etihad Stadium on Tuesday evening for a quarter-final tie which the home side will be expected to win by most observers.
However, Davies, who recently returned to action after injury, has called on his side to keep things as tight as possible and hope that they can then cause a shock later on in the game.
He told the Hull Daily Mail: "If we can spring a little upset, or at least go as far as we can in the game without conceding, then you know you're only one goal or a penalty away from getting through to the semi-finals.
"We'll go there as massive underdogs, but crazy things happen in football. A couple of years ago I don't think many people expected us to get to an FA Cup final, so let's keep our fingers crossed for another upset."
The Tigers can be backed at 7/1 to reach the semi-finals at the expense of their Premier League opponents, while they are 9/1 to win by a one-goal margin and 9/1to keep a clean sheet.
Meanwhile, Citizens head coach Manuel Pellegrini is expected to hand a start to fit-again Spanish playmaker David Silva, who came on as a late substitute in the 3-1 weekend Premier League win over Southampton.
Silva, who has recovered from an ankle problem, last began a game for City in early October, but while he is set to feature, Sergio Aguero will sit out after picking up a knock against the Saints.