The Ecuadorian defender has made 11 appearances in all competitions for United so far this season.
Ecuador defender Valencia, 31, has made 11 appearances in all competitions for United so far this season.
But he was missing from Jose Mourinho's squad for Saturday's league match at home to Burnley, with United confirming his injury via a Twitter feed statement that said: "Antonio Valencia has a fracture in his arm and will have surgery today."
Valencia, who made his name as a winger before featuring as a right back under successive United managers, has been replaced in the team to play Burnley by Matteo Darmian.
The 31-year-old Valencia's absence means he has now joined fellow defenders Eric Bailly and Chris Smalling in being sidelined through injury.
"They were the three players that start almost every match in the Premier League and now we lose all three in the same period," United manager Mourinho told broadcasters before Saturday's kick-off at Old Trafford.
"But that's football and it's an opportunity for others to play. We trust Darmian and we trust Marcos Rojo to have a positive influence."