The 22-year-old defender sustained the injury while on international duty earlier this week, having been forced off during the second half of England’s 2-0 win over Switzerland on Monday.
Jones had played every minute of United’s three matches in the Premier League this season as part of a new-look three-man backline implemented by manager Louis van Gaal.
The Dutchman is already light in defence, with Chris Smalling, Rafael (both groin) and Luke Shaw (hamstring) currently sidelined, although the latter two could make their returns this weekend against QPR.
Meanwhile, Marcos Rojo is also available to make his Red Devils debut on Sunday after finally being granted a visa following his off-season switch from Sporting CP.
But Van Gaal will have to do without Jones for the next three weeks at least after confirming the latest blow to his defensive options.
He said: “Phil has a torn hamstring. He is maybe out for three weeks or a month.”