Tarik Chadlioui, accused of being a recruiter for the Islamic State (ISIS) through extremist videos, was one of six people arrested across Europe in June as part of a Spanish investigation into support for the terrorist group.
Spanish authorities allege Chadlioui made three videos for extremist forces in Syria, during two visits to Mallorca in 2014 and 2015.
The 43-year-old faces a charge of collaboration with, or membership of, an armed group for which the maximum sentence is 20 years' imprisonment.
Chadlioui, a Belgian national of Middle Eastern origin, moved to the Sparkhill area of Birmingham in 2015, where he lives with his family comprising eight children.
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He is fighting against his extradition on the grounds that it will be "a clear interference with his right to a family life".
Spanish authorities believe he is behind the radicalisation of one of the Paris attack terrorists - Omar Mostefai - who killed 89 music fans at the Bataclan theatre in November, 2015, before blowing himself up.
Senior District Judge Emma Arbuthnot adjourned her ruling on the extradition case till Tuesday, until which time Chadlioui has been remanded in custody.