Real Sociedad scored once in each half to upset the European champions and leave them outside the top four at the halfway stage of the season - and struggling to qualify for nest season's Champions League.
Two goals did in Real as they fell to a 2-0 defeat at the Bernabeu, Willian Jose and Pardo netting in the first and last ten minutes for the visitors to pile the pressure on Solari.
And the former Argentina midfielder stubbornly refused to commit himself to questions about his future after the game.
"We compete to win the next game,” Solari told reporters when asked if he deserved the keep his position.
"We compete to win, always compete to win. We always fight to win. We never underestimate the competition, never underestimate anybody. The only one who has that luxury is he who has never competed.
“You can never question our winning character; we showed character today, despite the difficulties. Unforgivable would be to lack that.”
Real played for half an hour at the Bernabeu with ten men against Sociedad after Lucas Vasquez was sent off for two yellow cards, as they fell to a side that had only managed five league wins all season prior to this evening.
The result leaves Solari’s side seven points behind leaders and champions Barcelona, having played a game more.
The manager is widely expected to be replaced at the end of the season despite signing a two and half-year deal to replace former boss Julen Lopetegui in October, with Jose Mourinho and Roberto Martinez touted as possible replacements.
On the evening Solari admitted his side lost concentration, though they worked hard in their unsuccessful attempt to get back into the game.
"Nothing went right for us today," Solari said. "The sensation is that we wanted to make up three points and we didn't. This is football, we lost concentration at the start of the game that cost us a goal and hurt us. The same as in Villarreal [2-2 midweek], in professional football you cannot have that.
“Then we did everything possible to turn the result around, despite the difficulties, possible penalties, or penalties, and the sending off. We made chances but the ball just would not go in."