The results eliminated the club from the Champions League and the Copa del Rey, and ended its hopes of winning the Spanish league.
Solari was in charge of Real for 22 games in all competitions, losing eight and drawing two, and helped the Spanish giants claim the FIFA Club World Cup in December.
Zidane says there is work to be done to turn the club's fortunes around.
The French boss, Zidane, left Real at the end of last season and has been out of a job ever since. "First of all it's good to have [Zidane] back as a coach".
Despite Madrid's dominance in the Champions League, there is ground to make up in La Liga, where Barcelona are on course to seal their eighth league title in 11 years.
Solari led the team to a 4-1 win at Valladolid on Sunday and was in charge of practice on Monday. Ceballos had operated in several different positions in midfield, but Zidane could well opt to stick to tried and tested first-teamers Toni Kroos and Luca Modric.
As for Solari, after almost four and a half months in charge, he says goodbye to Madrid with a record of 21 victories, two draws and eight defeats.
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But when he left, Zidane hinted that he could return one day.
The club captain was also criticised for picking up a yellow card that ruled him out of the second leg of the last-16 tie, with UEFA deciding to ban him for an extra game after he suggested in a television interview that he was booked deliberately to avoid a possible quarter-final ban.
Zidane won an unprecedented three consecutive Champions League titles with the Spanish giants before resigning in May past year, with his replacement Julen Lopetegui sacked in October and Solari lasting just over four months in charge.
According to Marca, via The Mirror, Real Madrid are on the verge of sacking manager Santiago Solari and replacing him with Zidane. He lasted only a few months at the helm. Lopetegui and Solari wanted to do the best they could for the players, the team and the club. Unlike Zidane, though, he struggled to manage the squad and was at odds with some of its key players, including Marcelo and Francisco "Isco" Alarcon.
Asensio and Benzema did feature under Scolari, but the latter - without Cristiano Ronaldo's presence - is sure to flourish under the guidance of his countryman, while Asensio will be keen to show he can help improve Madrid's goal tally after the Portugal captain's absence.