The Portugal international was in action as UEFA Champions League holders Real drew 1-1 with Inter in the International Champions Cup in California on Saturday, before losing 3-2 on penalties.
Real are partaking in the pre-season tournament as they prepare for the new campaign.
Pepe revealed that coach Carlo Ancelotti is working the team hard in pre-season, but the 31-year-old accepts that will stand them in good stead for their quest for trophies in 2014-15 - which starts with the Super Cup against Liga rivals Sevilla on August 12.
"The truth is the coach demands a lot of us," he is quoted as saying by Marca.
"We're here to pick up the pace. We will be ready for the Super Cup. It's a trophy and we have a responsibility to everyone. We want to win it.
"Madrid, on the pitch, always has to win. We know that we have to go game by game and each game is very difficult, but we have to prove ourselves on the field."
Pepe also spoke of his excitement at playing in a side that will contain Gareth Bale, Cristiano Ronaldo and James Rodriguez.
"Gareth's had two weeks training and he looks great," he added. "He's got incredible ability and will be a very important this year.
"Cristiano had time to relax and unwind with his family. He's well and I think he'll be at his best.
On Rodriguez, who joined from Monaco for a reported fee of around €80 million last week, he added: "He's a very talented player. I saw him when he played at Porto and he has a very good left foot.
"It's amazing that Madrid have signed such a great player."
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