Barry is at Goodison Park on loan from Manchester City for the season and Martinez is impressed with the midfielder's attitude on and off the pitch.
The 32-year-old returns to his first club on Saturday when Everton take on Aston Villa, where he came through the ranks before a £12m move to the Etihad Stadium.
"It is not just in the games, it is in having him around," Martinez said.
"From the first day that he arrived he has been an incredible influence with the youngsters and an influence on the standards we have in training.
"Just with that alone you realise what a figure we have at the club. He is someone who breathes the game and brings the experience he has had from his career.
"He is playing with the desire and enthusiasm of an 18-year-old and has been a great player for us and I just want him to keep enjoying his football at Everton for as long as we can."
But Martinez would not be drawn on whether it was his intention to try and turn the move into a permanent one.
He added: "It is not the right time.
"I wouldn't put that in his mindset at the moment, all I want him to focus on is to keep enjoying his football and push for a start at the weekend and develop that competition behind the scenes - that is the only thing that matters at the moment."