PA PhotosReports in Italy suggest the US international has signed for a fee of 3.75 million euros.
Bradley, 24, has penned a contract that will keep him at the Stadio Olimpico until June 2016.
He has joined up with new Roma manager Zdenek Zeman and his side at their Brunico pre-season training camp.
The former Borussia Moenchengladbach midfielder has also spent time on loan at Aston Villa.
Bradley's father, Bob Bradley, is the Egypt national team coach and former USA boss.
It has also been reported that Chievo youngster Adrian Stoian has moved the other way as part of the deal.
The Flying Donkeys have received 50 per cent of the Romanian's rights in exchange for 500,000 euros, which is believed to be a reduced price on account of the Bradley deal.
