Based on Lawrence Wright's Pulitzer Prize-winning non-fiction book about the rise of Osama bin Laden and al Qaida in the late 1990s, the programme originally aired on Hulu.
The 10-part series follows counter-terrorism divisions at the FBI and CIA in the lead-up to the attack on the World Trade Centre on September 11 2001.
American actor and Dumb And Dumber star Jeff Daniels plays John O'Neill, the chief of the FBI's I-49 unit.
He and his protege, Muslim-American agent Ali Soufan, played by Tahar Rahim, are hindered in their investigations by their unco-operative counterparts in the CIA, led by Peter Sarsgaard's Martin Schmidt.
The series also guest stars Alec Baldwin as CIA director George Tenet.
Daniels was nominated in the lead actor in a limited series or movie category at the Emmys last year, alongside Benedict Cumberbatch for Patrick Melrose and John Legend for Jesus Christ Superstar.
BBC programme acquisition head Sue Deeks said: "Brilliantly weaving archive footage with tense and compelling drama, The Looming Tower is a fascinating and illuminating deconstruction of the personalities and events that ultimately led to the horror of 9/11."
The Looming Tower will air on BBC Two in spring.