The Spanish champions were ousted from Copa del Rey by third division Real Union, who went through to the last 16 on the away-goal rule after a 6-6 aggregate draw with Los Merengues.
Marca reported that the club's board held a temporary meeting after the match, and decided to give Schuster one last chance to stabilize the team, however, the German tactician sounded unfazed.
"I'm not worried about my job, but I understand that others will have some doubts. That's logical with what we are seeing," the 49-year-old said.
Real are currently fourth in the Primera Liga table, having conceded more goals than any other team in the top five.
Deteriorated Defense
Schuster admitted that his team are having a rearguard problem in the wake of the embarrassing cup exit.
"The problem is that teams score against us from the first chance they get," he told Real Madrid's official website.
"They don't dominate the match, they don't really put pressure on us, they simply score."
He showed great sportsmanship by lauding the performance of the third division side, whose qualification for the last 16 was an unexpected feat.
"I think Real Union should be congratulated because they played rather well in both games," he added.
"We weren't capable of adding tension to either match. You always have to go for it; talking about it before kick-off isn't enough."
On the other hand, he believes his players exerted substantial efforts in both games.
"I don't think there's an attitude problem; all my men ran and worked hard. The problem is that teams manage to get us into trouble with few chances on goal," he said.
Schuster's backline let in no less than 16 league goals, which is twice the goals-against record of leaders Barcelona.