The 19-year-old has found he has become a key figure in the new Toffees boss' plans for the style of play he wants to engender at Goodison Park.
Barkley has started every match for Everton this season and has been employed in an advanced role as a free-running midfielder.
It is a move which already appears to be paying dividends as Barkley has been able to pose more of a threat in and around opposition penalty areas and it was he who won the two free-kicks from which Leighton Baines scored in Saturday's 3-2 win at West Ham.
Everton return to London on Tuesday night for a League Cup tie at Fulham but, despite having played in the last round, Barkley may find Martinez wants to rest him.
"I'm young, I feel like playing every game and, for me, the more games the better," he told evertontv.
"It's been a good start to the season but I'm hoping to get better as it goes on."
Another player keen to make the most of Everton's unbeaten start to the season is on-loan Chelsea striker Romelu Lukaku, who scored on his debut at the weekend and will be fit to face the Cottagers, despite sustaining a head injury as a direct result of putting the ball in the net.
"He had that incredible impact on Saturday with the start of a new career at a new club and a new chapter in your football time," said Martinez.
"It is one of those good feelings you want to carry on and we are really pleased to have him in that frame of mind.
"The medical department are very happy with him so there are no concerns with Romelu."