The Argentinian has recently recovered from testicular cancer, and earlier this week thanked the Newcastle fans for their continued support.
Carver was appointed as the successor to Alan Pardew until the end of the season after the latter departed for Crystal Palace, and Gutierrez feels he should be afforded the opportunity to prove his worth in the role permanently.
"You always want the best for the manager," said Gutierrez.
"I've known John from five years ago, it's a good opportunity. I hope they [Carver and his coaching staff] can get it and stay permanent.
"All we can do is work hard and play well for them and get good results.
"If you do that, there's a lot of opportunities to stay in charge. We are working hard for them to keep the job."
Gutierrez also discussed making his first-team return, saying contractual issues make things complicated.
"I'm available as of a few weeks ago," he added. "I have a situation in my contract and things like that which means I'm not sure if I'm going to get the opportunity to play.
"I want to, but I know other things have to happen to be back. I have to work hard and be ready. If the situation changes I will do my best to get my opportunity."
