Brown, 31, has had a medical and is due to be unveiled on Thursday, while fellow defender O'Shea, 30, is also expected to agree terms.
A hitch over the move for Gibson, 23, is thought to centre on wage demands of around ?40,000 a week, but the Republic of Ireland international is ready to compromise.
Heading north: Wes Brown looks set to become Sunderland's latest signing
Brown headed for the north-east with Sunderland as they edged towards deals to secure their seventh and eighth summer transfer targets.
Significant progress has been made with both players in recent days, but the Republic of Ireland defender's situation has been complicated by the fact that his wife has given birth in the meantime.
Brown signed a first professional contract at Old Trafford within four months of joining the club as a trainee in 1996, and made his senior debut as a substitute against Leeds in May 1998.
Injuries have played a significant part in his career, but he formed a formidable central defensive partnership with Rio Ferdinand before Nemanja Vidic's arrival limited his opportunities.
Nevertheless, his time at United has been littered with winners' medals, including two in the Champions League.
Next up? John O'Shea (left) has also been targeted by Sunderland manager Steve Bruce
However, Sir Alex Ferguson's decision to accept Sunderland's bid suggested the time might be right to move on, and although he has also been linked with Everton and Bolton, the Black Cats have put themselves in pole position to push through a deal.
They are confident too of persuading O'Shea that his future lies at the Stadium of Light.
The 30-year-old utility man has made almost 300 starts for United and has five Barclays Premier League titles to his name, while he has become a stalwart of Giovanni Trapattoni's Ireland defence.
But while Brown and O'Shea may be close to joining, question marks remain over Gibson, who has also attracted interest from derby rivals Newcastle, with no agreement yet in place.
Bruce will hope for positive developments over the next 24 hours after a busy summer already, with Ahmed Elmohamady having made his loan move permanent and Connor Wickham, Craig Gardner, Ji Dong-won, Keiren Westwood and Sebastian Larsson having joined the club.
Big spender: Steve Bruce has been busy in the transfer market since the season ended
The players are back in pre-season training and Bruce is hoping that his squad will be boosted further by the time they play their first friendly at York next Wednesday.
But there was one notable absence with 28-year-old Paraguay international midfielder Cristian Riveros having completed a 12-month loan deal with Turkish club Kayserispor.
Riveros has struggled to make an impression since his free transfer move from Mexican club Cruz Azul, and has made just 14 appearances.
Bruce said: 'Cristian wants to play regular football and hopefully he will be able to do so in Turkey. We wish him all the best.'
Sunderland aim to secure United trio as Bruce bids to continue spending spreeSunderland lead the chase for Manchester United defender BrownRiveros leaves the Stadium of Light but Gyan is going nowhere, say SunderlandAll the latest Sunderland news, features and opinion