The Argentine wants his compatriot to commit his future to the club after recovering from a bad start to his career in England to become a key player for Alan Pardew's side
The Magpies captain is out of contract at the end of next season and has been linked with a move away from the club, with Liverpool said to be interested earlier this campaign.
Gutierrez also praised Coloccini, who won the 2011 north-east football of the year award, and the rest of his Newcastle team-mates' performances in a season which has seen them mount a serious challenge for a Champions League spot.
He told the Sunday Sun: "I am so glad that he is here with our club and yes, of course I would like him to stay and I said that to him.
"But all of the players are having a great season. It is not just about him - we're all having a good season.
"That is what this season is all about."
Gutierrez went on to hail the defender's improvement since moving to Newcastle from Deportivo La Coruna in 2008.
"I have known Fabricio for a number of years, I know exactly what kind of player he is," he added.
"Maybe in the first season he was bad. But it was bad for all of the players who arrived here because we were relegated.
"But I know the qualities that he has. He always tries to play the quality pass and he's a really, really good defender too."