Howard has made over 300 appearances for the Toffees, but Martinez believes that new arrival Joel Robles will "fight" the American for the number one jersey.
The 40-year-old told Evertontv: "As you know, since Jan Mucha has moved on, Tim Howard was the only goalkeeper at the football club.
"It was important to get another goalkeeper with the quality of Joel on a permanent basis. He has experience in the Premier League and was part of the squad that won the European Under-21 Championships with Spain.
"He is someone who is going to give us great protection and fight with Tim Howard for the goalkeeping position and push him all the way."
Robles has moved to Everton from Atletico Madrid, but he did play 13 matches during a loan spell with Wigan last season.