According to an ESPN report, the 42-year-old Solari was handling the club's youth side Castilla, but took over the first team after Julen Lopetegui was sacked after the Los Blancos 5-1 bashing in El Clasico by Barcelona.
Solari has led Real to four wins out of four since being installed on a caretaker basis following Lopetegui's abrupt sacking last month.
The exciting attacker has also scored in a Champions League final.
I don't rate his chances of being charge beyond the end of this season I have to be honest.
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"It is an inconsistent Barcelona by their own high standards".
"Great for him, delighted for him, but I wouldn't view it as long-term employment".
News that he had been rewarded with a permanent contract surfaced on Monday when the Spanish federation announced: "Everything is in order". Following the announcement, Solari said, "I'm going to go into things positively and do everything I can".
Real Madrid are understood to be leading the race for his signature and would be able to sign him for just €7.5m (£6.5m) should they activate a buy-back clause which was included in the initial deal to take him to Espanyol a year ago.
"Everything goes very quickly, the important thing is the day to day".