The Mirror says has sounded out his former Manchester United Treble-winning team-mate about a six-month loan switch to the Stadium of Light.
Beckham has played just five games for new club LA Galaxy, who invested £128million in his signing.
Keane will face a battle to prize Beckham away from the United States given Galaxy's investment. But the chance to stay fit and in the spotlight for McClaren could prove a temptation.
Black Cats boss Keane is trying to sell him the dream of being a new messiah in the North East by turning out in front of the passionate Sunderland fans.
Their former United pals Dwight Yorke and Andy Cole have already been snapped up by Keane, who believes Becks could help his inexperienced side climb away from the relegation zone.