They would reach the position for the first time since they won the league in 2012.
Ever since the start of the season Real Madrid have been playing catch up with defending champions Barcelona, and now neighbours Atletico.
At one stage in December Carlo Ancelotti's men trailed the leading two by five points, but Los Blancos are now within touching distance of the summit.
Madrid moved just one point off the top last weekend when they beat Real Betis 5-0 while Barca and Atletico could only draw in their respective fixtures against Levante and Sevilla.
This weekend Ancelotti's side take on Granada on Saturday afternoon, more than 24 hours before Barca face Malaga and Diego Simeone's team travel to Rayo Vallecano.
Everyone at the Santiago Bernabeu knows the importance of Saturday afternoon's game.
"We have to take advantage of this moment and win to put a bit of pressure on those above us," Madrid defender Alvaro Arbeloa said in AS.
All the omens are looking good for Ancelotti's side ahead of the clash.
Madrid have won their last six matches without conceding a goal and they have only lost once at home this term - back in September to Atletico.
Gareth Bale scored his first goal for six weeks in a 5-0 rout of Real Betis last weekend and on Wednesday they secured a 1-0 win over Espanyol in the Copa Del Rey.
With Atletico only managing a draw this week, and Barca dealing with the fall out from Sandro Rossell's resignation, there is a feeling that Real Madrid's time is now.
"We are on a great run, we have cut the gap to the top bit by bit," Arbeloa added.
"We have hit form at the most important time of the season."
Diego Costa's goals have helped Atletico form a serious title challenge, but there is no doubt about who Arbeloa views as Madrid's main rivals.
The defender said: "Atletico are doing extraordinarily well, but we all know that the enemy for us is Barcelona."
Granada have lost 5-1 and 3-0 in their last two visits to the Bernabeu, but their away record has been good this term.
The Andalucian side have lost just three times on the road and they sit 10th in the table.
For coach Lucas Alcaraz, a Granada man who started his coaching career at the club 19 years ago, taking his hometown team to the Bernabeu for the first time is something he will savour.
"You do pinch yourself a little bit," said Alcaraz, who replaced Juan Antonio Anquela as coach 12 months ago.
"I have to thanks those at the club.
"The fact that Granada have been going to the Bernabeu for the last three years is very important for the club and you have to thank (president) Quique Pina and (director) Juan Carlos Cordero.
"These are the people who rescued Granada and made it possible for the club to play out these scenarios. Now this is normal for the club.
"If we play our best we can take points from there."
Alcaraz has new signing Tiago Ilori available after he joined on loan from Liverpool.