Barry was one of three arrivals at Everton on deadline day after the club had concluded deals for Wigan's James McCarthy and Chelsea's Romelu Lukaku earlier in the day.
And the 32-year-old is looking forward to playing under Martinez as well as the Goodison Park faithful after joining on a season-long loan.
"The way I've seen his teams play over the years as soon as he got his first management job in English football he's always tried to play the right style of football," said Barry.
"When I've played against his teams it's always been interesting to see what formations he's played.
"He's made it difficult for the teams playing against him and the way he had plans to fit me into the squad was very encouraging - the manager had a big part to play."
He continued: "I've enjoyed coming to Goodison Park and playing in great atmospheres.
"The fans here are fantastic and they always make it tough for opposing teams so it will be great to be playing on their side."
Barry was a mainstay in the City midfield under Roberto Mancini and formed a key part of the side that were crowned Premier League champions in 2012.
The England international is keen to help Everton improve and hinted he may be open to the possibility of extending his stay on Merseyside.
"One target for this season is to improve on last season and obviously things don't happen overnight," said Barry.
"The club and the fans are aware that it is a new manager and his ideas may take a bit of time to get across but it's been a good start for the club and I'm sure we can improve on that.
"They've always been there or thereabouts. That's why I wanted to come here and help the team and help the club improve - certainly this year and even after that."