Real set a Spanish record for consecutive victories (22) in all competitions at the end of 2014 but stuttered after the turn of the year and coach Carlo Ancelotti was sacked after they failed to win the Champions League, La Liga or King's Cup.
Barca, who won all three competitions last term, have been in scintillating form in recent weeks and hammered Real 4-0 at the Bernabeu last weekend before demolishing AS Roma 6-1 in Champions League Group E on Tuesday.
They entertain Real Sociedad in La Liga on Saturday when a win will put them seven points clear of second-placed Atletico Madrid who host Espanyol later in the day.
Real, two points behind Atletico in third, visit sixth-placed Eibar on Sunday.
"I have never been worried about that," Luis Enrique told a news conference on Friday when asked if he thought Barca were peaking too early.
"The idea is to play well for the whole season because obviously the titles are won at the end," added the former Barca and Spain midfielder.
"It's great to play like that against our eternal rival last weekend but we have to be able to do it as well at the key moments.
"We are playing well but there is a long way to go, two thirds of the season."
Barca will have midfielder Arda Turan and versatile full back Aleix Vidal available in January when a FIFA ban on registering new players expires.