Barca became the first, and so far only, Spanish side to achieve the feat in 2009 and with coach Luis Enrique in his first year at the helm are well placed for a repeat performance with five matches left in La Liga.
Through to the Champions League last four to face Bayern Munich, Barca will seek a record extending 27th King's Cup triumph when they play Athletic Bilbao in next month's final and have a two-point lead over Real at the top of the domestic standings.
Real, who have been depleted by injuries in recent weeks, battled to a 4-2 comeback victory at Celta Vigo on Sunday after Barca eased to a 2-0 success at neighbours Espanyol on Saturday.
In a busy week before the Champions League resumes, Barca host Getafe on Tuesday and Real are at home to Almeria on Wednesday before the leaders play at bottom side Cordoba and Real face a tough match at Sevilla on Saturday.
"We're going to treat every game like a cup final and give our all in every one of them," Real fullback Marcelo told reporters after Sunday's win at Celta, when the European champions had to come back from a goal down.
"The team is in good shape and on incredible form," added the Brazil international.
"We're confident, we can't relax. We're going to be in there fighting.
"Our aim is really clear in our minds and we're going to give everything right up until the last moment."
Real have what is possibly the tougher run-in to the end of the campaign, with their trip to Sevilla followed by a game at home to Valencia, who can climb above the Andalusians into fourth if they avoid defeat at home to relegation-threatened Granada later on Monday.
Barca's trickiest remaining game is on the penultimate matchday at champions Atletico Madrid, who are seven points behind Real in third ahead of their game at sixth-placed Villarreal on Wednesday.

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