The England international sat out Saturday's 1-0 victory against Tottenham, after which United confirmed that he is set for a longer spell on the sidelines.
The club wrote on Twitter: "Phil Jones, who was not involved in today's matchday squad, has mild thrombosis and will be out for a minimum of three weeks."
Jones's injury will not only see him miss United's Champions League play-off tie with Club Brugge but he must also be considered doubtful for England's European Championship qualifying double against San Marino and Switzerland at the start of September.
In his absence, Daley Blind played alongside Chris Smalling in the heart of Louis van Gaal's defence against Spurs.
