Solari led Real's reserve team from 2016 until he took over from Lopetegui.
Odegaard is on-loan at Vitesse from Real this season - and showing his best form since signing for Madrid.
Argentinian Solari made almost 150 appearances for Real as a player in the early 2000s and returned to the club in 2013 as youth coach.
After seeing off Melilla, Real Valladolid and Viktoria Plzen, the Argentinian oversaw a 4-2 defeat of Celta Vigo which was more than enough to convince Florentino Perez in handing him the permanent manager's job.
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, who spent five years playing for Real, was promoted from his role in charge of Castilla, the club´s B team, after Lopetegui had overseen a miserable run that included just one victory in seven games. "We are very happy with how things have gone".
Spanish rules state that an interim coach can not be in charge for more than two weeks so Madrid had to appoint Solari - or an alternative - to avoid punishment.
"It's a great opportunity and it's a great club", Solari said.