The Academy product has progressed quickly this season and secured his first senior international call in August before earning his first cap when coming on as a sub in the 4-0 win over Moldova.
Barkley, 19, has made 11 Premier League appearances for Everton so far this season and Martinez has been happy to expose him to top-flight opposition to speed up his progress through the ranks.
"Ross has been very successful in any challenge that has come his way in his football career," Martinez told Sky Sports News.
"It has never been a doubt how well he is going to embrace whatever comes his way."
Martinez, who was showing members of the United States rugby league team around their Finch Farm training complex, has his sights firmly set on his first Merseyside derby in charge against Liverpool next weekend.
He added: "We are both in a very good position after a good start to the season.
"We are both playing with confidence and with a smile on our faces, and that's the perfect platform to see a good game of football. We hope it is going to go our way."