According to Reuters and AFP, the Spanish Football Federation have received Solari's full-time contract, which means he will continue at the helm until the end of the season.
Solari could not have had a much better start to the job - in the four games he has overseen, the team has posted four wins and only conceded in yesterday's game against Celta.
After Julen Lopetegui was sacked following an embarrassing 5-1 defeat to Barcelona in the clásico, Solari was made interim boss and has taken Madrid on a run of four consecutive wins, the latest of which came on Sunday night at Balaídos away to Celta Vigo.
Madrid are now sixth in La Liga, four points behind leaders Barcelona, who lost 4-3 to Real Betis at the Camp Nou on Sunday.
Though the club have yet to confirm the news, Spanish rules prevent an interim coach from staying in charge for longer than 15 days, a period that expires on Monday.
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Solari also heaped praise on Karim Benzema, who is so often the scapegoat whenever Madrid are struggling but who has been in fine form, having scored twice in the 5-0 drubbing of Viktoria Plzen in the Champions League in midweek.
"The important thing is to do what is necessary and possible for the good of the team, for the good of Real Madrid", Solari said.
'Celta played very aggresively, ' Solari said. "I don't just say that now as coach of the first team".
'We Italian coaches have a great culture of work and when you join a new club you can't change everything, you must try and do it very gradually. But the one thing Solari has managed to do is put his name up for the job, with the stars at Real Madrid giving the former Castilla manager a glowing recommendation.
Real Madrid's first match back after the worldwide break is away at Eibar on November 24.
Madrid were looking for a big name following the sacking of Lopetegui with Antonio Conte and Mauricio Pochettino the favourites.