Both teams issued statements on Monday, the day after the Italian Grand Prix at Monza.
Hadjar missed that race and the Dutch Grand Prix before it after he suffered a wrist injury while training last month.
"The decision has been taken as Isack's recovery continues to progress positively," said Red Bull, who had four-time world champion Max Verstappen finish third at Monza.
"While he is not yet ready to return to the cockpit, he will be in Madrid supporting the team."
Tsunoda has stepped up from his role as reserve driver and the Japanese will partner British rookie Arvid Lindblad at Racing Bulls.
The Madring is making its Formula 1 debut this weekend as the new home of the Spanish Grand Prix, previously hosted by Barcelona's Circuit de Catalunya.