The Magpies face Russian side Anzhi Makhachkala in the first leg of their last-16 tie in the Europa League this week with Alan Pardew admitting they have as good a chance as anyone left in the competition to lift the trophy.
However, Gutierrez has stressed that Premier League survival must remain the club's top aim after Saturday's defeat to Swansea left them just six points above the drop zone.
"From the beginning of the season the most important thing has been the Premier League," said Gutierrez. "The Europa League is a nice competition for us but the first ambition is the Premier League. That is our priority.
"I have been in Spain and in England and I have seen big teams playing in the Champions League but who are in the bottom three. We must think game-by-game but we need to start taking the points to make ourselves feel comfortable in the league for the rest of the season.
"Of the last five games, we've won three. We lost at Tottenham, who are having a good season, and then on Saturday against Swansea we didn't deserve to lose. These things happen, we have to keep going.
"Ask the fans of Newcastle and the people of Newcastle if they want to win the Europa League and get relegated I think they will say no.
"The most important thing for Newcastle is the Premier League. Of course we will do our best in the Europa League but we need to get the points to be in a comfortable position in the league too."