Joel Robles is set to deputise for the suspended Tim Howard against Southampton, while Gareth Barry is also absent, so the Spaniard wants his fringe players to impress
The Merseyside outfit head into the fixture with suspensions to goalkeeper Tim Howard and midfielder Gareth Barry, with the former having been sent off in Thursday's 1-0 defeat to Sunderland while the latter picked up his fifth booking of the season in the same game.
Joel Robles is expected to fill in between the posts, while Martinez could rotate his midfield due to a hectic end-of-year schedule, and the Spaniard believes now is the perfect time for the whole squad to stake a claim for a starting spot.
"You get to this moment of the season, the amount of games you play in a short period of time, and December is where you can accumulate a fifth booking, and you pick up injuries," the former Wigan boss told reporters.
"December and January, for whatever reason, you get injuries, you get tested.
"Tim Howard has been sensational for us, setting high standards on a regular basis. Joel is a determined young man and wants to press for a first-team spot. We couldn't be happier with the goalkeepers at the club.
"At the same time, we're missing Gareth Barry, a player who has set really high standards. We will miss him but it's a chance for someone else.
"We will never use an excuse not to perform. It's an opportunity for the players that haven't been playing to show how hard they're working behind the scenes."
Everton's defeat to Sunderland on Boxing Day ended their unbeaten home record in 2013 and was just their second loss in the Premier League this season.
But Martinez was happy with how his team responded to losing Howard early on, and believes there are positives to take from defeat.
He added: "What we can take [from the Sunderland game] is the way we faced adversity. We lost a man early on and had to play 60 minutes with 10 men. We did that magnificently. I'm happy with the performance.
"There's a real sense of responsibility as a team, we don't want to find an excuse, we always look at ways we can overcome the difficulties and perform in a manner that can give us the result."