No financial details were given but British media reported a fee of nine-million pounds ($12.89 million) had been agreed.
French-born Khazri, an attacking midfielder, is Bordeaux's top scorer in the league this season with five goals.
Sunderland, second from bottom in the league, also signed Ivory Coast central defender Lamine Kone from FC Lorient this week.
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During the transfer window the north-east club also recruited Senegal striker Dame N'Doye on loan from Turkey's Trabzonspor and defender Jan Kirchhoff from Bayern Munich.
Speaking of his latest capture, Sunderland manager Sam Allardyce said of the 24-year-old Khazri: "He is a quick, dynamic attacking player."
(Reporting by Martyn Herman; editing by Justin Palmer)
