The 26-year-old USA international is on Paolo Di Canio’s list of targets and would cost upwards of £6m.
A number of clubs are understood to be interested in Bradley, who has 80 caps for USA.
But after failing to land Tom Huddlestone, who joined Hull City, Sunderland are understood to have cast their net wider.
Bradley scored one goal and created two for Roma last season after joining from Chievo in 2012.
He had a short spell on loan at Aston Villa in 2011, when he was contracted to German side Borussia Monchengladbach.
The New Jersey-born midfielder would be Sunderland’s second American signing of the summer, after Jozy Altidore joined in July.
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