Lopetegui, 52, was clinging to his job after a 2-1 defeat at home to Levante on Saturday.
The former Spain boss replaced Zinedine Zidane in the summer but Real are fifth in the La Liga table after a dreadful start to the season.
And Spanish media outlet Marca claim they are ready to wield the axe before their Champions League match with Viktoria Plzen at the Bernabeu on Tuesday.
Real then face a crunch clash with Barcelona in El Classico next Sunday.
And it is reported that club chiefs want to act to prevent a potentially embarrassing defeat to their bitter rivals.
Reports claim Real Madrid B boss Solari is the man they will turn to, although senior players are said to be unhappy at the prospect of him taking the top job.
While Solari, 42, has a fine pedigree as a player, his record as a manager is unconvincing and reports claim Real’s biggest stars want the club to remain patient with Lopetegui.
The Argentine has taken charge of the second team for 42 games – with 14 wins, 13 draws and 15 defeats.
As a player, he won the Champions League with Real in 2001-02 and lifted the La Liga title twice.
He went on to win Serie A three times with Inter Milan as well as the Coppa Italia.
However, It is said the players are not convinced he is ready to take a job of such magnitude and Ok Dario claim they are right behind Lopetegui.
Lopetegui has endured a torrid year. he was sacked by Spain just days before the start of the World Cup after he announced he would join Real when the tournament had finished.
And he appears to be on the brink of a second sacking in a matter of months.