The 30-year-old Republic of Ireland international was on his way to Wearside on Thursday morning hot on the heels of team-mate Brown, who was in the North-East a matter of hours earlier.
A successful completion of both deals would take manager Steve Bruce's summer signings to date to eight.
Bruce targeted the pair, as well as midfielder Darron Gibson, several weeks ago when the Black Cats had a joint bid of more than £12million accepted for the three.
Gibson was quick to indicate his willingness to consider a move to the Stadium of Light, but that deal has proved to be the most problematic.
However, first 31-year-old Brown and now O'Shea have edged ever closer to accepting Sunderland's offer to add experience and know-how to a defence which was badly affected by injuries last season.
Both men are able to play in more than one position and have enjoyed success during long stays at Old Trafford, and that winning mentality is something Bruce is keen to bring to his squad.
He has already secured a permanent deal for winger Ahmed Elmohamady, the only one of his loan signings so far to agree a permanent switch, while the club snapped up Connor Wickham, Craig Gardner, Ji Dong-won and free signings Keiren Westwood and Sebastian Larsson in a flurry of transfer activity ahead of Wednesday's return for pre-season training.