Manchester United will face Sunderland at lunchtime on Saturday and victory would extend their lead at the top of the table to 18 points by the time Newcastle face City at the Etihad.
But Gutierrez insists that Newcastle's motivation is purely selfish as they battle to climb clear of danger at the other end of the table.
He told the Chronicle: "We don't care about their dreams - we care about our position and our jobs.
"We know that we have to take something. If it's a draw, then great - if we can win, brilliant. But we want to get a point to make up our tally.
"City are 15 points behind with 24 points left to play for. When you are Man City or Man United, you aren't going to drop many points.
"And for Man City they have to just keep doing their best. Man United have one hand already on the Premier League trophy - I don't think they'll lose.
"But in football maybe Man City can still gain 24 points, they are good enough. Man City have a lot of top players.
"We hope it's not their day - even Carlos Tevez has a bad day and the others are also human.
"If we are focused and play well we can take advantage. All of the teams in the Premier League are difficult.
"We saw in the last game against Wigan that they were bottom three but they played a great game.
"We deserved something out of it but we got beat. Maybe in the next game we'll deserve to lose and take three points. That's football. If we concentrate for 90 minutes we should get something."
The match will come too soon for Tim Krul but the keeper could be back for next week's Europa League match with Benfica.
He damaged ligaments last month against Metalist Kharkiv but returned to light training on Monday.
He tweeted: "Completed my first full training session today with the lads. Buzzing to be back on the pitch. Proper freezing out there."