The 32-year-old was one of three arrivals at Everton on deadline day after the club also concluded deals for James McCarthy and Romelu Lukaku.
Barry was allowed to leave Manchester City following their summer recruitment drive - and Martinez admits he would like to see the former Aston Villa man stay on Merseyside for "many years to come".
The Spaniard told Everton's official website: "What we want is to find a way where Gareth can come and be a part of Everton and enjoy his football, and we can all enjoy his experience and his quality.
"After that, we will see if there is an opportunity to see him for many years to come - because I do believe Gareth has got a long-term future at the club."
Barry also appears open to joining Everton on a permanent basis - but knows he must deliver the goods during his season-long loan spell first.
"Obviously that will happen if I can produce good performances this season, stay fit and put a good run of games together," he said when introduced to the media at Finch Farm on Tuesday.
"That is all down to me impressing the club, so I don't see why that can't happen."
Barry's current deal at parent club City expires next summer, and earlier on Tuesday he revealed that the chance to play under Martinez was a big factor in his decision to join the Toffees.
"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 the midfielder.
"When I've played against his teams it's always been interesting to see what formations he's played.
"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.
"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."