Busquets left the team's pre-Euro 2020 training camp on Sunday, while the rest of the squad tested negative after their friendly draw with Portugal in Madrid at the weekend.
"The players are getting daily tests right now, the Spanish federation said.
Enrique has drafted in Leeds striker Rodrigo Moreno, midfielders Pablo Fornals of West Ham and Carlos Soler of Valencia plus Brais Mendez of Celta Vigo.
They will train away from the rest of the players and at different times.
Spain continue their warm up for the Euros with a match against Lithuania on Tuesday.
Their first Euro 2020 match is against Sweden on June 14 and Enrique has until Saturday to make changes to his squad ahead of their opening fixture in Seville.