The European champions made it 15 wins in a row for their third time in history with a 1-0 UEFA Champions League success against Basel on Wednesday.
Miguel Munoz's men achieved the feat in 1961, and so did Jose Mourinho's side three years ago.
Neither team were able to win a 16th time in a row, though, meaning more history is up for grabs for Carlo Ancelotti's side.
"We are group leaders, the team have been great," Ancelotti said after the Basel win. "The players deserve appreciation for what they're doing.
"It's a fantastic streak, but the season is very long and you have to think about the next game.
"We have to think about improving this winning streak.
"Anything we can do to get ourselves in the club's history is a good thing.
"A huge amount of credit must go to the players. They are having a fantastic season."
Real Madrid are a club soaked in history and for Ancelotti's men to be on the brink of a record speaks volumes of the work done by the Italian.
It is incredible to think that Ancelotti's job security was being questioned following a pair of Liga defeats in a row just two months ago.
After losing 4-2 at Real Sociedad, last year's Copa del Rey winners went down 2-1 at home to Atletico Madrid - who they were also beaten by in the Supercopa de Espana.
Ancelotti's decision to let Angel di Maria and Xabi Alonso leave the club was questioned but the critics have since been silenced by a breathtaking run of wins - in which they have scored an incredible 57 goals and conceded just eight.
Talisman Cristiano Ronaldo scored the sole goal in Real's Champions League win in Switzerland and the Portuguese star has an outstanding record against Malaga.
He has scored 13 in his last 10 league matches against the club and it would surprise no-one if he added to that record at La Rosaleda.
Real sit two points clear of rivals Barcelona on top of La Liga but also have reigning champions Atletico Madrid snapping at their heels four points adrift.
Malaga are in good form too, though, and had a five-match winning streak end at Atletico last week.
They lost that match 3-1, a clash that saw both sides finish with 10 men.
Malaga sit sixth in the league but have scored just 15 goals - 31 less than Real.
Luka Modric (thigh), Asier Illarramendi (groin) and Jese Rodriguez (knee) are set to be Real's only absentees.