The rift began in December 2006 when Southgate accused Cristiano Ronaldo of diving after he won a penalty in United's 2-1 victory in the league.
Three months later, Ronaldo won another penalty in the dying moments of an FA Cup sixth round replay between the two teams.
Another Cristiano Ronaldo 'Dive' Could Spark It Off Again!
Middlesbrough's James Morrison was then sent off for a wild lunge on the Portuguese winger and Southgate appeared to condone his player's conduct.
In the Carling Cup this season, Emmanuel Pogatetz saw red for a dreadful foul on United's Rodrigo Possebon and later apologised for his challenge.
Sir Alex Ferguson: "We hope the Christmas spirit has lingered on long enough to help heal the rift that had developed between the two clubs.
"I am sure we can build a better relationship. I certainly hope so because we have had some momentous games against Middlesbrough, matches that have helped decide titles.
"I certainly admire the work of Gareth Southgate, a young manager who made the bold and brave decision to change the shape of his team, moving on older players to make way for a fresh, young squad.
"I cannot help but admire his ambition and conviction."