Apple Inc.’s 13 billion-euro ($14.4 billion) battle with the European Union reaches the bloc’s courts next month in a hearing set to throw the spotlight on antitrust commissioner Margrethe Vestager’s crackdown on tax deals doled out to big companies.

The EU’s General Court, its second-highest tribunal, will hear arguments in the challenges by the iPhone maker and Ireland over two days set for Sept. 17-18. The U.S. last year lost a bid to intervene in the case in support of Apple.

The European Commission in August 2016 ordered Ireland to recoup the record sum plus interest, saying the world’s richest...