"He's gone to the hospital, so we'll just have to assess it," Giggs said.
"We'll see how it looks, but it didn't look great."
Jones, 22, hurt himself in an aerial collision with Maynor Figueroa mid-way through the first half of the Premier League game at Old Trafford and had to be helped off the pitch by a member of United's medical staff.
The news will be of particular concern to England manager Roy Hodgson, who is due to name his 23-man squad for the World Cup in Brazil on Monday.
The versatile Jones has won nine caps for England and was expected to be included in Hodgson's squad.
Source: AFP