Hart is rated as one of the best goalkeepers in the world and has been the first choice between the posts for England since the 2010 World Cup finals.
"He reminds me of Casillas," Martyn told talkSPORT. "He got into the team at Real Madrid very early and you can see the career he has had.
"Hopefully Joe can emulate that because he's doing fantastically well. As long as you steer clear of the injuries, don't do anything silly and stay at one of the top clubs then it all bodes well for his international future.
"It's up to the even younger lads to try and oust him and that will make him better as well."
Hart will feature for England against Ukraine on Tuesday in their second game of the 2014 World Cup qualifying campaign.