League rules mandate that an interim coach can not stay longer than a fortnight in the post, meaning Madrid's hierarchy must either confirm Solari as their permanent coach on Monday or look elsewhere for a new full-time boss.
Interim boss Solari oversaw a fourth consecutive win since succeeding the sacked Julen Lopetegui last month - and could now be handed the Real reins full-time.
Brazil worldwide midfielder Casemiro was forced off inside the opening 20 minutes of the Blancos' 4-2 win over Celta Vigo on Sunday, with the Santiago Bernabeu outfit confirming he has suffered a sprained ankle.
With his temporary spell coming to an end, Solari may yet get the Madrid job permanently, and Butragueno said he was delighted with the Argentinian.
"The most important thing is to do whatever is necessary for the good of the team, nothing else matters", he told reporters when asked if he expected to be given the job.
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But Solari was upset with forceful Celta challenges, which often went unpunished.
Real Madrid's number one goalkeeper made a fine save to keep out a strike from Aspas in the 69th minute as Celta chased a second goal against Solari's team, who lost Nacho to injury in the 71st minute as their defensive problems continued.
"Their challenges were at the limit of what's allowed and we suffered because of that".
Real Madrid director Emilio Butragueno praised Santiago Solari, saying the club were "very happy" with the caretaker coach.
"It was a very complicated game and we also had setbacks with injuries, [but] we had personality and character". Celta tried to press and counter their makeshift backline but were met with resistance in the form of Sergio Ramos and Casemiro, who instantly launched the ball forward in the search for a goal.
'I have seen the play on television and there is no penalty from Junca to Odriozola and a very clear one from Lucas Vazquez to Emre Mor that was not noted, ' he said, as quoted by Spanish media outlet Marca. "We are very happy with how things have gone".