The 24-year-old has entered Del Bosque's thinking ahead of next week's World Cup qualifying games against Belarus and Georgia after scoring eight goals in eight Primera Division matches for Atleti in the new season.
Costa, who originally joined Atletico in 2007 and returned to Madrid from Real Valladolid three years ago, was granted Spanish nationality in July and the Royal Spanish Football Federation (RFEF) last month asked FIFA for permission to register the player.
He has, however, represented Brazil in two friendly matches this year - against Italy and Russia in March - and FIFA has now written back to the RFEF demanding "complementary requisites" before approving his registration.
The game's world governing body has also asked the Brazilian Football Confederation to provide a certificate confirming Costa has not played in an "official match" for his country of birth.
A statement published by RFEF on Thursday read: "The Atletico Madrid player, Diego Costa, will not be able to be called up tomorrow by the coach Vicente del Bosque for the following national team games against Belarus and Georgia, due to FIFA requiring complementary requisites.
"This procedure is still in progress, even though the player has verbally manifested his intention of playing with the Spanish national team to the Spanish coach.
"In relation to the eligibility of the player Diego da Silva Costa to play in teams representative of the Royal Spanish Football Federation, FIFA has provided a written answer which states that it is a general rule for the association which looks to call up a player, has full responsibility for him to play in one of the representative teams, to test whether the player in question is eligible or not at least to play in teams of the association."
Del Bosque will name a squad of players to face Belarus and Georgia in Group I on October 11 and 15 respectively and is short of an experienced striker after Chelsea's Fernando Torres was ruled out for three weeks through injury.
Spain are tied for top spot with France but Les Bleus have only one game - against Finland on October 15 - left to play.