The 22-year-old sustained the injury during England's 2-0 win over Switzerland in Basle on Monday, and had to be replaced by Phil Jagielka.
In a press conference on Friday, United boss Louis van Gaal confirmed that Jones would miss Sunday's Premier League clash with QPR, while also highlighting concerns over a number of other players.
"[Marouane] Fellaini is more than a month away [following a ankle injury]," said Van Gaal. "Jones maybe three weeks or a month.
"[Michael] Carrick [who was ruled out with an ankle problem prior to the season starting] isn't ready but has run for the first time on grass."
Van Gaal also revealed that left-back Luke Shaw "is fit enough to play some minutes" but bemoaned United's lengthy injury list.
"[Sam] Johnstone has elbow problems, [Chris] Smalling isn't fit enough, [James] Wilson and [Reece] James are not ready yet," added the Dutchman.
"[Ashley] Young and [Jesse] Lingard are also injured."
Van Gaal is still waiting for his first competitive win as United boss, with his side having taken just two points from three Premier League games.
United also suffered an embarrassing 4-0 defeat to MK Dons in the second round of the League Cup last month.
