Nike has been supplying United's famous red shirts since 2002 but the current agreement expires next year. Its withdrawal is expected to clear the way for the club to sign a lucrative new deal with German sportswear company Adidas.
"Clearly Nike's football business is booming and they could easily afford to do this (deal). They had the right to match any other offer but looking at the deal on the table, they will walk away as it doesn't make commercial sense," the source said.
United got around 38 million pounds ($64.7 million) from the Nike deal in 2012-13, including its share of profit from the sale of merchandise around the world.
Media reports had suggested the club was looking for a new deal worth around 60 million pounds per season.
Nike and Adidas are battling for supremacy in the soccer kit industry, with the American company having made big inroads into a sport that its German rival long dominated.
United, owned by the American Glazer family, are looking to rebuild after a disappointing season in which the 20-times English champions finished only seventh in the Premier League.
The club has installed Dutchman Louis van Gaal as their new manager and he will take charge after the World Cup where he has steered the Netherlands to the semi-finals.
Adidas and Man Utd both declined to comment.