The 31-year-old Argentinian, whose contract expires at the end of a difficult campaign, was part of the team which dropped into the Championship in 2009 after a final-day defeat at Aston Villa.
Six years on, Gutierrez and his team-mates know only victory over West Ham on Sunday - they have not won in 10 matches since February 28 - will guarantee their continued presence in the top-flight, although they will be safe anyway if Hull fail to beat Manchester United at the KC Stadium.
The South American told the Journal: "I don't have another option than to just think about the game. It's the most important game I have had to play in seven years.
"The good thing is that we play at home and it's in our hands. A few players here know what it's like to go down, and it's not nice for the city, the players and the club.
"We need to move forward to the next game and do it for the fans and the city."
Gutierrez has found himself plunged into the thick of a relegation battle after winning his personal fight against cancer, and while Sunday's game at St James' Park could prove emotional for a variety of reasons, he insists only the club's fate is important.
He said: "The fans have been really good with me since I arrived, and through my illness. I really appreciated their support.
"I try to do my best for the team and for them. That is more important."