The 32-year-old came through the ranks at Villa before moving to Manchester City and now plies his trade at Goodison Park after a transfer deadline-day loan agreement was struck.
Everton boss Martinez, who also brought in Chelsea's Romelu Lukaku on a temporary deal, reckons Barry has been a great individual to have in and around the club.
"It is not just in the games, it is in having him around," he 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."
Despite being happy with Barry's start at the club, Martinez refused to be drawn on whether he would be looking to make the England international's move a permanent one.
"It is not the right time," he added.
"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."
One player whose long-term future at Goodison Park also appears to remain in the balance is that of Leighton Baines.
The 28-year-old Baines has started the season in great form for club and country but Martinez confirmed no negotiations have opened with the former Wigan man, who has under two years remaining on his current contract.
Asked about a new deal for Baines, Martinez replied: "There is no train of thought on that at the moment.
"At the moment all that matters is to see Leighton fully fit and enjoying his football. All his performance stats have improved from last season and that is important to me.
"That is what matters, everything else is just trying to find an issue from the outside to see what we can comment on."
Moyes' United side currently sit one place and four points below Everton in the Barclays Premier League table with the Toffees just one point off a place in the top four.
Martinez feels it would need a monumental effort from his players to end the season with Champions League qualification but he reckons his ever-improving side are in with a shout.
"I think it [the close table] shows that there are six clubs that all are expected to be fighting for the Champions League spots," he said.
"Anyone who wants to break into those top four positions it is going to take an important effort.
"I look at the six points we dropped earlier in the season with the three draws where we were really good in everything we did and six points now would make a big difference in the table.
"In the manner that we have been consistent game after game allows me to think that the squad is ready to be pushed, challenge each other and is ready to hit good heights and that is what we are going to do.
"Game by game we will become better, become stronger and when we get around April, March we will know where we are and what we can fight for in the league."
Fit again Steven Pienaar will be challenging for a start after scoring just seconds after coming off the bench in last weekend's win against Hull.
Martinez also confirmed Darron Gibson has responded well to surgery on a cruciate knee ligament injury and expects defender Antolin Alcaraz to be available after the next international break.