The 19-year-old has established himself as one of the first names on the Toffees' teamsheet this season, and his new boss believes he can handle the added pressure.
The 19-year-old Barkley has started all six of the Toffees’ Premier League games so far, with his impressive performances earning him an international debut in September and a place in Roy Hodgson’s latest England squad.
The Capital One Cup defeat against Fulham was the only Everton match Barkley has not been involved in this term, and Martinez believes the youngster is relishing his new-found responsibility.
"Ross is upset even when he can't win in training never mind when he can't be involved in games," Martinez told Sky Sports. "He is a young man who has been waiting for this opportunity for a long time and is desperate to enjoy every single second.
"When you are young you don't feel tired. He has a terrific energy and it is just the start of what is going to be a very special season for him."
Martinez has been particularly impressed with how quickly Barkley has gelled with Chelsea loanee Romelu Lukaku, after the pair linked up impressively in Monday’s 3-2 win over Newcastle.
"You see good players always understand each other,” Martinez said. “I thought Ross and Romelu clicked in the second half of the West Ham game and have developed in training a bit of a natural understanding of each other's game. You are talking about 20-year-old players who excite you for the future.
"Their potential is terrific and we have to keep working and developing them because they are an important part of our team now. You can always work on certain aspects of the player but normally it is a natural relationship that is allowed to develop.
"Good players always get to know each other well in the way they enjoy their football. It is something which they can grow and improve in the future."