The 42-year-old Argentine has led the team to four wins in four games since replacing Julen Lopetegui at the helm of the European champions two weeks ago.
NAN reports that the Spanish rules prevent an interim coach from staying in charge for longer than 15 days, which expire on Monday (today), without being appointed on a permanent basis.
The Spanish Football Association, RFEF, told ESPN that Madrid had filed papers to hire Solari on that basis.
Solari has enjoyed the best start of any manager in the club’s history with victories over Melilla, Viktoria Plzen, Real Valladolid and Celta Vigo.
Lopetegui was sacked after a poor run of form that saw Madrid sink to ninth in the table.
After Sunday’s 4-2 win at Celta, the capital club are currently sixth, four points behind league leaders, Barcelona.