Forward Costa, who plays his club football with Atletico Madrid, has been struggling with a series of niggles in recent weeks and was forced off in the ninth minute of Saturday's Champions League final against Real Madrid.
"He has a muscle injury and we will monitor his development so we can make a last-minute decision," Del Bosque said on Spanish radio late on Monday.
"We have until June 2 to register the (final) squad of 23 and we don't need to rush," he added. "I hope he makes it but we have good replacements."
If Brazil-born Costa is unable to make the trip to his native country for the finals starting next month, Del Bosque has several players available with a similar profile, including Fernando Torres, Alvaro Negredo and Fernando Llorente.
Spain play a World Cup warmup against Bolivia in Seville on Friday, after which the Real and Atletico players selected for the finals will join up with the squad.