Madrid took a giant step towards the semi-finals of the Champions League on Wednesday when they beat Borussia Dortmund 3-0 at the Bernabeu.
Carlo Ancelotti was brought in last summer to end Madrid's long wait for their 10th European Cup, but the Italian is well aware that domestic success is also required by one of the most demanding boards in world football.
Madrid looked near certainties for the title when they embarked upon a strong run at the turn of the year, but back-to-back defeats to Barcelona and Sevilla mean Los Blancos are now three points behind leaders Atletico Madrid, who lead second-placed Barca by one point.
Only six games remain, but Alonso is refusing to give up on the title just yet.
"In the league we have lost the advantage that we had, but we need to keep fighting because there could be surprises," Alonso told his club's official website.
"The teams that beat us were in great form, we didn't play well in two games and we committed errors that we paid very heavily for."
The one bit of bad news to emerge from the quarter-final first leg win over last year's beaten finalists was the early withdrawal of Cristiano Ronaldo due to a knee injury.
The club later played down the scale of the injury, and Alonso revealed the Portuguese star is on course for a swift return.
"Cristiano did the right thing in coming off as a precaution, but I don't think he will have any problems in playing on Saturday," Alonso said.
"On Thursday he trained normally with the team."
With Barcelona and Atletico both playing their respective games against Real Betis and Villarreal early in the day, Madrid could be six points off the summit by the time they face Real Sociedad on Saturday evening.
Barcelona and Athletic Bilbao both lost at the Anoeta Stadium earlier this year and Jagoba Arrasate's men still harbour hopes of bridging the six-point gap that separates them and fourth-placed Athletic.
Having spent five years at Sociedad, Alonso knows how much pride the Basque side have in their home record.
"Playing in Anoeta is always difficult," Alonso said.
"They have had two seasons playing well and enjoying great results so the fans are thrilled with the team.
"It will be a difficult game and we can't do anything other than win."
Sociedad received a boost going into the match after Xavi Prieto returned to training following an injury which ruled him out of last Sunday's disappointing 1-1 draw at Osasuna.
Fellow midfielder Inigo Martinez says his team-mates must respect Saturday's star-studded opposition.
"It will be difficult. They have players who are very quick and can punish you," he told Real Sociedad's website.
"We have to be attentive.
"They are a big team, but we are playing at home, where we have been strong this year and we will fight right up until the last minute."