Brown underwent a medical for Steve Bruce’s club yesterday, while fellow Manchester United defender O’Shea will do so today.
Bruce reached a joint deal of around £12m last month for the defenders plus Red Devils midfielder Darron Gibson.
While the defenders’ moves are near completion, Gibson’s has stalled due to disagreements over personal terms.
Brown, 31, and 30-year-old O'Shea would mark the seventh and eighth Black Cats signings of the season.
It was reported yesterday that Sunderland midfielder Cristian Riveros will spend a season’s loan spell at Turkish club Kayserispor.